LINKING FORMS, SINGULARITIES, AND
HOMOLOGICAL STABILITY FOR DIFFEOMORPHISM
GROUPS OF ODD DIMENSIONAL MANIFOLDS
by
NATHAN GIRARD PERLMUTTER
A DISSERTATION
Presented to the Department of Mathematics
and the Graduate School of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2015
DISSERTATION APPROVAL PAGE
Student: Nathan Girard Perlmutter
Title: Linking Forms, Singularities, and Homological Stability for Diffeomorphism
Groups of Odd Dimensional Manifolds
This dissertation has been accepted and approved in partial fulfillment of the
requirements for the Doctor of Philosophy degree in the Department of Mathematics
by:
Dr. Boris Botvinnik Chair
Dr. Hal Sadofsky Core Member
Dr. Yuan Xu Core Member
Dr. Peng Lu Core Member
Dr. Spencer Chang Institutional Representative
and
Dr. Scott L. Pratt Dean of the Graduate School
Original approval signatures are on file with the University of Oregon Graduate
School.
Degree awarded June 2015
ii
c© 2015 Nathan Girard Perlmutter
iii
DISSERTATION ABSTRACT
Nathan Girard Perlmutter
Doctor of Philosophy
Department of Mathematics
June 2015
Title: Linking Forms, Singularities, and Homological Stability for Diffeomorphism
Groups of Odd Dimensional Manifolds
Let n ≥ 2. We prove a homological stability theorem for the diffeomorphism
groups of (4n + 1)-dimensional manifolds, with respect to forming the connected
sum with (2n − 1)-connected, (4n + 1)-dimensional manifolds that are stably
parallelizable. Our techniques involve the study of the action of the diffeomorphism
group of a manifold M on the linking form associated to the homology groups of
M . In order to study this action we construct a geometric model for the linking
form using the intersections of embedded and immersed Z/k-manifolds. In addition
to our main homological stability theorem, we prove several results regarding
disjunction for embeddings and immersions of Z/k-manifolds that could be of
independent interest.
iv
CURRICULUM VITAE
NAME OF AUTHOR: Nathan Girard Perlmutter
GRADUATE AND UNDERGRADUATE SCHOOLS ATTENDED:
University of Oregon, Eugene, OR
DEGREES AWARDED:
Doctor of Philosophy, Mathematics, 2015, University of Oregon
Master of Science, Mathematics, 2012, University of Oregon
Bachelor of Science, Mathematics and Philosophy, 2009, University of Oregon
AREAS OF SPECIAL INTEREST:
Algebraic Topology, Singularity Theory, Moduli Spaces of Manifolds
PROFESSIONAL EXPERIENCE:
Graduate Teaching Fellow, University of Oregon, September 2010- June 2015
GRANTS, AWARDS AND HONORS:
Anderson Grad. Research Award, University of Oregon Dept. of Math., 2014
D.K. Harrison Memorial Award, University of Oregon Dept. of Math., 2013
Rose Hills foundation scholarship, Rose Hills foundation, 2012 and 2015.
PUBLICATIONS:
N. Perlmutter, The linking form and stabilization for diffeomorphism groups
of odd dimensional manifolds. Submitted: arXiv:1411.2330
N. Perlmutter, Cobordism Categories of Manifolds With Baas-Sullivan
Singularities, Part 2. Submitted: arXiv:1306.4045
N. Perlmutter, Cobordism Categories of Manifolds With Baas-Sullivan
Singularities, Part 1. To appear in Mu¨nster Journal of Mathematics. (2015)
N. Perlmutter, Homological stability for the moduli Spaces of products of
spheres. To appear in Trans. Amer. Math. Soc. (2015)
v
ACKNOWLEDGEMENTS
I thank Boris Botvinnik, my thesis advisor, for his unwavering support,
encouragement, and for teaching me what doing real Math is all about. I also
thank my parents for their support in all of my creative and intellectual pursuits.
vi
TABLE OF CONTENTS
Chapter Page
I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Background and Basic Definitions . . . . . . . . . . . . . . . . . . 2
1.2. Some Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3. Statement of Our Main Results . . . . . . . . . . . . . . . . . . . . 9
1.4. Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5. Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II. SIMPLICIAL TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . 18
2.1. Cohen-Macaulay Complexes . . . . . . . . . . . . . . . . . . . . . 18
2.2. Topological Flag Complexes . . . . . . . . . . . . . . . . . . . . . 24
2.3. Transitive Group Actions . . . . . . . . . . . . . . . . . . . . . . . 26
III. LINKING FORMS AND ODD-DIMENSIONAL MANIFOLDS . . . . . . 29
3.1. Linking Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.2. The Homological Linking Form . . . . . . . . . . . . . . . . . . . . 32
3.3. The Classification Theorem . . . . . . . . . . . . . . . . . . . . . . 33
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Chapter Page
IV. THE PRIMARY COMPLEXES . . . . . . . . . . . . . . . . . . . . . . . 36
4.1. The Complex of Connected-Sum Decompositions . . . . . . . . . . 36
4.2. The Complex of Linking Forms . . . . . . . . . . . . . . . . . . . . 39
4.3. High-Connectivity of the Linking Complex . . . . . . . . . . . . . 41
V. Z/K-MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1. Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2. Bordism of 〈k〉-Manifolds . . . . . . . . . . . . . . . . . . . . . . . 50
5.3. Z/k-Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4. Immersions and Embeddings of 〈k〉-Manifolds . . . . . . . . . . . . 53
VI. 〈K,L〉-MANIFOLDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.1. Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6.2. Oriented 〈k, l〉-Bordism . . . . . . . . . . . . . . . . . . . . . . . . 57
6.3. 1-dimensional, Closed, Oriented, 〈k, k〉-Manifolds . . . . . . . . . . 58
VII. INTERSECTION THEORY OF Z/K-MANIFOLDS . . . . . . . . . . . . 60
7.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
7.2. Intersections of 〈k〉-Manifolds . . . . . . . . . . . . . . . . . . . . . 60
7.3. 〈k, l〉-Manifolds and Intersections . . . . . . . . . . . . . . . . . . . 62
7.4. Main Disjunction Theorem . . . . . . . . . . . . . . . . . . . . . . 64
viii
Chapter Page
7.5. Connection to the Linking Form . . . . . . . . . . . . . . . . . . . 66
VIII. HIGH CONNECTIVITY OF |X•(M,A)K | . . . . . . . . . . . . . . . . . 68
8.1. The Complex of 〈k〉-Embeddings . . . . . . . . . . . . . . . . . . . 68
8.2. A Modification of K(M)k . . . . . . . . . . . . . . . . . . . . . . . 74
8.3. Reconstructing Embeddings . . . . . . . . . . . . . . . . . . . . . . 77
8.4. Comparison with X•(M,a)k . . . . . . . . . . . . . . . . . . . . . 83
IX. HOMOLOGICAL STABILITY . . . . . . . . . . . . . . . . . . . . . . . . 87
9.1. A Model for BDiff∂(M) . . . . . . . . . . . . . . . . . . . . . . . . 87
9.2. A Semi-Simplicial Resolution . . . . . . . . . . . . . . . . . . . . . 89
9.3. Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . 90
X. EMBEDDINGS AND DISJUNCTION FOR Z/K-MANIFOLDS . . . . . 95
10.1. A Z/k Version of the Whitney Trick . . . . . . . . . . . . . . . . 95
10.2. A Higher Dimensional Intersection Invariant . . . . . . . . . . . . 100
10.3. Creating Intersections . . . . . . . . . . . . . . . . . . . . . . . . 104
10.4. A Technical Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 105
10.5. Modifying Intersections . . . . . . . . . . . . . . . . . . . . . . . . 107
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Chapter Page
XI. IMMERSIONS AND EMBEDDINGS OF Z/K-MANIFOLDS . . . . . . . 114
11.1. A Recollection of Smale-Hirsch Theory . . . . . . . . . . . . . . . 114
11.2. The Space of 〈k〉-Immersions . . . . . . . . . . . . . . . . . . . . . 115
11.3. Representing Classes of 〈k〉-Maps by 〈k〉-Immersions . . . . . . . 118
11.4. The Self-Intersections of a 〈k〉-Immersion . . . . . . . . . . . . . . 119
11.5. Modifying Self-Intersections . . . . . . . . . . . . . . . . . . . . . 121
11.6. Representing 〈k〉-Maps by 〈k〉-Embeddings . . . . . . . . . . . . . 126
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
x
CHAPTER I
INTRODUCTION
Let M be a smooth manifold and let Diff(M) denote the group of
diffeomorphisms of M topologized in the C∞-topology. The classifying space,
BDiff(M), occupies a central place in smooth topology. In particular, for any
paracompact space X there is a natural bijection between the set of homotopy
classes of maps X → BDiff(M) and the set of isomorphism classes of principal
Diff(M)-bundles over X. This bijection reduces the study of principal Diff(M)-
bundles to the study of the weak homotopy type of the classifying space BDiff(M).
Furthermore, the cohomology ring H∗(BDiff(M)) contains all characteristic classes
for principal Diff(M)-bundles.
For an arbitrary manifold M , the space BDiff(M) is quite complicated and
there are very few “universal” results that pertain the homotopy type of BDiff(M)
for all manifolds M . A fundamental question is the following: to what extent does
the homology group H∗(BDiff(M)) depend on the structure of the underlying
manifold M? In particular, how does the structure of the group H∗(BDiff(M))
change if we alter the underlying manifold M by a surgery operation? Recently,
through the work of M. Weiss and I. Madsen in (28), and the work of S. Galatius
and O. Randal-Williams (11), (10), much progress has been made in answering
this question when the manifold M is of dimension 2n. This thesis is primarily
concerned with answering this question in the case that the manifold M is of
dimension 2n+ 1.
1
1.1. Background and Basic Definitions
Fix a smooth manifold M . Let Diff∂(M) denote the subgroup of Diff(M)
consisting of all diffeomorphisms of M that restrict to the identity on some
neighborhood of the boundary ∂M . Our main object of interest is the classifying
space of the topological group Diff∂(M). For any topological group G, the
classifying space BG is characterized up to weak homotopy equivalence as the
base space of a principal G-bundle EG −→ BG with weakly contractible total
space EG. We will need to work with a particular model for the classifying space of
Diff∂(M) which we define below.
Definition 1.1.1. Fix a collar embedding h : (−∞, 0] × ∂M −→ M with
h−1(∂M) = {0} × ∂M , and fix an embedding θ : ∂M −→ {0} × R∞. We define
EDiff∂(M) to be the space of all smooth embeddings φ : M −→ (−∞, 0] × R∞,
for which there exists a real number  > 0 such that φ(h(t, x)) = (t, θ(x)) when
− < t ≤ 0. We then define BDiff∂(M) to be the quotient of EDiff∂(M) obtained
by identifying embeddings that have the same image.
With the above definition, the underlying set of BDiff∂(M) is precisely the
set of all submanifolds V ⊂ (−∞, 0] × R∞ that are abstractly diffeomorphic to
the manifold M and that have boundary equal to θ(∂M). We will generally denote
elements of BDiff∂(M) by such submanifolds V . In this way BDiff∂(M) can be
thought as a “non-linear” generalization of a Grassmannian manifold.
Remark 1.1.1. The definition of BDiff∂(M) involved two arbitrary choices: the
collar h and the embedding θ. Any two embeddings ∂M ↪→ R∞ are isotopic.
Similarly, any two collar embeddings (−∞, 0] × ∂M −→ M are isotopic as well.
It follows that the topology of the space EDiff∂(M), and hence the topology of the
2
space BDiff∂(M), does not depend on the choices of embedding θ and h used in the
definition. For this reason we are justified in excluding h and θ from the notation.
The topological group Diff∂(M) acts freely and continuously on EDiff∂(M)
by pre-composition (f, φ) 7→ φ ◦ f , and thus BDiff∂(M) is identified with the
orbit space EDiff∂(M)/BDiff∂(M). By (4), it follows that the quotient projection
EDiff∂(M) −→ BDiff∂(M) is a locally trivial fibre-budle with fibre equal to
Diff∂(M). Furthermore, it is well known that the total space EDiff∂(M) is weakly
contractible. It follows that our definition of BDiff∂(M) given in Definition 1.1.1
coincides with the standard definition of the classifying space BG for a general
topological group G given in (32). It then follows that for any paracompact
space X, there is natural bijection between the set of homotopy classes of maps
X −→ BDiff∂(M) and the set of isomorphism classes of principal Diff∂(M) fibre
bundles over X.
We will need to work with certain natural maps connecting the classifying
spaces of the diffeomorphism groups of different manifolds. Let M be a smooth
manifold of dimension m with non-empty boundary. Let K be an m-dimensional
manifold with boundary given by the disjoint union ∂K = ∂0K unionsq ∂1K, where
∂0K = ∂M (in other words, K is a cobordism from ∂M to ∂1K). Let M ∪∂M K
denote the manifold obtained by attaching K to M along ∂M .
Definition 1.1.2. Let M and K be as above. Choose a collared embedding α :
K ↪→ [0, 1]×R∞ with α(∂iK) ⊂ {i}×R∞ for i = 0, 1, and such that α|∂0K coincides
with the embedding ∂M ↪→ {0} × R∞ used in the definition of BDiff∂(M). Let
t−1 : (−∞, 1]×R∞ −→ (−∞, 0]×R∞ be the diffeomorphism given by translation in
3
the first coordinate. We define
∪K : BDiff∂(M) −→ BDiff∂(M ∪∂M K)
to be the continuous map given by sending an element V ∈ BDiff∂(M) (which is
a submanifold of (−∞, 0] × R∞) to the element of BDiff∂(M ∩∂M K) given by the
submanifold t−1(V ∪ α(K)).
Remark 1.1.2. There was an arbitrary choice involved in the construction of
the map in the above definition, namely the embedding α. Since any two collared
embeddings K ↪→ [0, 1] × R∞ are isotopic, it follows that the homotopy class of the
map ∪ K in the above definition does not depend on the choice of embedding
α. It follows that the homotopy class of ∪ K is determined entirely by the
diffeomorphism class of the cobordism K.
A particular case of the above map that we will use is the following. Let W
be a closed manifold of dimension m and let KW denote the manifold obtained
by forming the connected sum of ∂M × [0, 1] with W . We identify the manifold
M ∪∂M KW with the connected-sum M#W and thus the classifying spaces
BDiff∂(M ∪∂M KW ) are BDiff∂(M#W ) identified. Using these identifications, the
map from Definition 1.1.2 yields a map
sW : BDiff
∂(M) −→ BDiff∂(M#W ). (1.1)
1.2. Some Fundamental Results
Perhaps the first positive result pertaining to the homomorphism on
homology induced by the map sW : BDiff
∂(M) −→ BDiff∂(M#W ), is the
4
homological stability theorem of J. Harer in (13) for the diffeomorphism groups
of oriented surfaces. For an integer g ∈ N, let Lg denote the oriented surface
with boundary obtained by removing an open disk from the closed genus-g surface
(S1 × S1)#g. Since Lg+1 ∼= Lg#(S1 × S1), for each integer g (1.1) yields the map
s : BDiff∂(Lg) −→ BDiff∂(Lg+1). The main theorem from (13) is the following.
Theorem 1.2.1 (J. Harer 1985). The map on homology
s∗ : H`(BDiff
∂(Lg);Z) −→ H`(BDiff∂(Lg+1);Z)
induced by s is an isomorphism when ` < g
3
− 2.
The above theorem implies that the maps in the direct system
· · · // BDiff∂(Lg−1) // BDiff∂(Lg) // BDiff∂(Lg+1) // · · · (1.2)
induce isomorphisms on the homology groups H`( ;Z) when ` << g. We say then
that the direct system (1.3) satisfies homological stability and that Theorem 1.2.3
is a homological stability theorem. Significant improvements in the stability range
of the above theorem were made years latter, separately by Ivanov (18), Boldsen
(6), and Randal-Williams (31). Furthermore, an analogue of Theorem 1.2.1 for non-
orientable surfaces was proven by N. Wahl in (40).
Remark 1.2.1. In (8) it was proven that the diffeomorphism groups Diff∂(Lg) have
contractible path components when g > 1. It follows that BDiff∂(Lg) is homotopy
equivalent to the classifying space of the mapping-class group pi0(Diff
∂(Lg)) when
g > 1. The proof of Theorem 1.2.1 makes use of these facts and for this reason,
5
Theorem 1.2.1 is usually stated in terms of the mapping-class groups of surfaces,
rather than of the diffeomorphism groups.
The above theorem has many formal similarities to other classical homological
stability theorems regarding direct systems of groups. For example, for large classes
of rings R, the direct system of classifying spaces of the general linear groups,
BGln(R) → BGln+1(R) → BGln+2(R) → · · · , is known to satisfy homological
stability (see (20)). Also the direct system, BΣn → BΣn+1 → BΣn+2 → · · · of
classifying spaces of the symmetric groups satisfies homological stability as well (see
(29)). The proof of Theorem 1.2.1 shares many formal similarities with the proofs
of these above mentioned homological stability results.
In (28) I. Madsen and M. Weiss identify the homology of the limiting space
colim
g→∞
BDiff∂(Lg). Let CP∞−1 denote the Thom-spectrum associated to the formal
inverse of the canonical complex line bundle γ1C over CP∞. We will need to
consider the infinite loopspace Ω∞CP∞−1 associated to the spectrum CP∞−1. We will
denote by Ω∞0 CP∞−1 the path-component of Ω∞CP∞−1 containing the constant loop.
The following is the main theorem from (28).
Theorem 1.2.2 (Madsen-Weiss 2002). There is a homological equivalence,
colim
g→∞
BDiff∂(Lg)
∼=−→ Ω∞0 CP∞−1.
The cohomology of the space Ω∞0 CP∞−1 can be readily computed and thus
the above theorem, together with Theorem 1.2.3, enables a calculation of the
cohomology (and homology) groups of the space BDiff∂(L2ng ) in a dimensional
range when g is large. With rational coefficients there is an isomorphism
H∗(Ω∞0 CP∞−1;Q) ∼= Q[κ1, κ2, . . . ], where κi ∈ H2i(Ω∞0 CP∞−1;Q) for i ≥ 0. The
6
classes κi are known as the kappa classes and were studied by Miller, Morita, and
Mumford in (25), (26), and (27). This theorem of Madsen and Weiss proved the
long-standing Mumford Conjecture from (27), which conjectured the isomorphism
H∗(Ω∞0 CP∞−1;Q) ∼= Q[κ1, κ2, . . . , ].
Remark 1.2.2. If one views the symmetric group Σg as the diffeomorphism group
of the zero-dimensional manifold consisting of g-many points, then the theorem
of Barratt, Priddy, Quillen and Segal, establishing the homological equivalence
colim
g→∞
BΣg×Z '−→ QS0 (see (2)), can be viewed as an analogue of the Madsen-Weiss
theorem for zero-dimensional manifolds. The proof of theorem 1.2.4 has formal
similarities to the theorem of Barrat, Priddy, Quillen, and Segal as well.
We now discuss some recent results pertaining to the diffeomorphism groups
of high dimensional manifolds analogous to those from the previous section. Let
n ∈ N be an integer an let M be a manifold of dimension 2n with non-empty
boundary. We will consider the map s : BDiff∂(M) −→ BDiff∂(M#(Sn × Sn))
and the direct system
· · · // BDiff∂(M#(Sn × Sn)#g) // BDiff∂(M#(Sn × Sn)#(g+1)) // · · ·
(1.3)
obtained by iterating this map s.
The idea of studying manifolds of dimension 2n up to connected sum with
factors of Sn × Sn can be traced back to Wall in his study of simply connected 4-
manifolds in (38). This idea was taken further by M. Kreck in (22) where he gives
a diffeomorphism classification of manifolds of dimension 2n ≥ 6, up to connected
sum with factors of Sn × Sn.
7
In order to state the main theorems regarding the direct system (1.3), we will
need to define a generalized notion of genus for manifolds of dimension 2n. For
each g ∈ N, let L2ng denote the manifold obtained by deleting an open disk from
(Sn × Sn)#g. We define g(M) to be the largest integer g ∈ N such that there exists
an embedding of L2ng into M . The following result is the main theorem from (11).
Theorem 1.2.3 (Galatius, Randal-Williams 2014). Let M be a simply connected,
compact manifold of dimension 2n ≥ 6, with non-empty boundary. Let g(M) ≥ g.
Then the map on integral homology
s∗ : H`(BDiff
∂(M);Z) −→ H`(BDiff∂(M#(Sn × Sn));Z)
induced by s is an isomorphism when ` ≤ 1
2
(g − 3) and an epimorphism when
` ≤ 1
2
(g − 1).
The above theorem implies that for any simply connected manifold M of
dimension 2n ≥ 6 with ∂M 6= ∅, the maps in the direct system (1.3) induce
isomorphisms on the homology groups H`( ;Z) when g >> `. Thus the above
theorem is a homological stability theorem analogous to Theorem 1.2.1 from the
previous section.
In addition to the homological stability theorem described in the previous
paragraphs, in (10) the authors identify the homological type of the limiting space
colim
g→∞
BDiff∂(L2ng ). Let γ2n −→ BO(2n) denote the universal 2n-dimensional vector
bundle, let θn : Bθn −→ BO(2n) denote the n-connective cover, and let MTθn
denote the Thom-spectrum associated to the formal inverse of the pull-back bundle
θ∗n(γ2n) −→ Bθn.
8
Theorem 1.2.4 (Galatius, Randal-Williams 2014). Let n ≥ 3. Then there is a
homological equivalence, colim
g→∞
BDiff∂(L2ng )
'−→ Ω∞0 MTθn.
The cohomology of the space Ω∞0 MTθn can be readily computed (it has
been completely determined rationally) and thus the above theorem, together
with Theorem 1.2.3, enables a calculation of the homology groups of the space
BDiff∂(L2ng ) in a dimensional range when g is large. The proof of Theorem 1.2.4
was largely inspired by the proof of the Madsen-Weiss theorem and uses many of
the same techniques.
1.3. Statement of Our Main Results
The results about the diffeomorphism groups discussed in the previous
sections all pertain solely to manifolds of even dimension. Indeed, the results
of Galatius and Randal-Williams from (11) rely heavily on techniques from the
surgery theory of Kreck (22) and Wall (35) specialized to the category of manifolds
of dimension 2n ≥ 6. The homological type of the diffeomorphism groups of
manifolds of odd dimension have until now been largely untouched and there are
very few existing results in the literature. In this thesis we prove a homological
stability theorem for the diffeomorphism groups of manifolds of odd dimension. We
extend the techniques used by Galatius and Randal-Williams to the category of odd
dimensional manifolds, and thus also set the stage to approach an odd dimensional
version of Theorem 1.2.4.
Let M be a (4n + 1)-dimensional manifold with non-empty boundary, where
n ≥ 2. For any closed (4n + 1)-dimensional manifold W , we consider the map
9
BDiff∂(M) −→ BDiff∂(M#W ) from (1.1) and the direct system
BDiff∂(M) −→ BDiff∂(M#W ) −→ · · · −→ BDiff∂(M#W#g) −→ · · · (1.4)
obtained by iteration of this map. The main result of this thesis is a theorem
concerning the homological stability of the above direct system.
Theorem 1.3.1. Let n ≥ 2 and let M be a 2-connected, (4n + 1)-dimensional,
compact manifold with non-empty boundary. Let W be a closed, (4n + 1)-
dimensional manifold that satisfies the following conditions:
– W is (2n− 1)-connected,
– W is stably parallelizable,
– the homology group H2n(W ;Z) has no 2-torsion.
Then the group H`(BDiff
∂(M#W#g);Z) is independent of the integer g if g ≥
2`+ 3. In particular, the direct system (1.4) satisfies homological stability.
Remark 1.3.1. The case of Theorem 1.3.1 when W is a product of spheres
S2n × S2n+1 follows as a special case of (30, Theorem 1.3) by the same author of
this thesis.
The proof of the above theorem draws heavily from the classification of
(2n − 1)-connected, (4n + 1)-dimensional manifolds of Wall in (37). In order
to describe the techniques used in the proof of Theorem 1.3.1, we must describe
this classification theorem of Wall. Let us first fix some notation that we will use
throughout the paper. Let W4n+1 denote the set of all (2n− 1)-connected, (4n+ 1)-
dimensional, compact manifolds. Let W¯4n+1 ⊂ W4n+1 denote the subset of those
10
manifolds that are closed, let WS4n+1 ⊂ W4n+1 denote the subset of those manifolds
that are stably-parallelizable, and let W¯S4n+1 denote the intersection WS4n+1 ∩ W¯4n+1.
In order to prove Theorem 1.3.1, we will need to analyze the diffeomorphism
invariants associated to elements of W4n+1. For M ∈ W4n+1, let piτ2n(M) ≤ pi2n(M)
denote the torsion subgroup. The primary diffeomorphism invariant associated to
M is the linking form, which is a skew-symmetric, bilinear pairing
b : piτ2n(M)⊗ piτ2n(M) −→ Q/Z, (1.5)
which is non-singular in the case that M is closed. For n ≥ 2, the classification of
manifolds in W4n+1 was studied by Wall in (37). Recall that two closed manifolds
M1 and M2 are said to be almost diffeomorphic if there exists a homotopy sphere
Σ such that M1#Σ is diffeomorphic to M2. It follows from Wall’s classification
theorem (37, Theorem 7), that two elements M1,M2 ∈ W¯S4n+1 are almost
diffeomorphic if and only if there exists an isomorphism, piτ2n(M1)
∼=−→ piτ2n(M2)
that preserves the linking form b. Furthermore, given any finite abelian group G
equipped with a non-singular, skew-symmetric bilinear form b′ : G ⊗ G −→ Q/Z,
there exists a manifold M ∈ W¯S4n+1 and an isomorphism of forms, (piτ2n(M), b) ∼=
(G, b′).
We use the classification result discussed above to specify certain elements of
W¯S4n+1. For each integer k ≥ 2, fix a manifold Wk ∈ W¯S4n+1 whose linking-form
(piτ2n(Wk), b) is given by the data,
pi2n(Wk) = Z/k ⊕ Z/k, b(σ, σ) = b(ρ, ρ) = 0, b(σ, ρ) = −b(ρ, σ) = 1k mod 1,
(1.6)
11
where 〈ρ, σ〉 is the standard basis for Z/k ⊕ Z/k. It follows from (37, Theorem 7)
and the classification of skew symmetric bilinear forms over Q/Z in (38, Lemma 7),
that any element M ∈ W¯S4n+1 is diffeomorphic to a manifold of the form
Wk1# · · ·#Wkl#(S2n × S2n+1)#g#Σ, (1.7)
where Σ is a homotopy sphere.
Remark 1.3.2. It follows from these classification results, (37, Theorem 7) and
(38, Lemma 7), that if k and ` are relatively prime, then Wk#W` ∼= Wk·`. In this
way, the (almost) diffeomorphism classification of W¯S4n+1 mirrors the classification
of finitely generated abelian groups. Thus it will suffice to restrict our attention to
the manifolds Wk in the case that k = p
j for a prime number p. By the connected-
sum decomposition (1.7) it follows that the manifolds Wpj are indecomposable.
Now, let M be a (4n + 1)-dimensional manifold with non-empty boundary.
For each integer k ≥ 2, let
sk : BDiff
∂(M) −→ BDiff∂(M#Wk) (1.8)
denote the map from (1.1). We will refer to this map as the k-th stabilization map.
Let rk(M) be the quantity defined by,
rk(M) = max{g ∈ N | there exists an embedding, W#gk \D4n+1 −→M}. (1.9)
For each k we may think of rk(M) as a generalized version of the genus of a surface
or an analogue of the quantity g(M) used in the statement of Theorem 1.2.3. Using
the diffeomorphism classification for manifolds in W¯S4n+1 described in Section III,
12
the following result, combined with (30) implies Theorem 1.3.1. This is the main
homological stability result that we prove in this paper.
Theorem 1.3.2. For n ≥ 2, let M be a 2-connected, compact, (4n+ 1)-dimensional
manifold with non-empty boundary. If k > 2 is an odd integer, then the map on
homology induced by (1.8),
(sk)∗ : H`(BDiff
∂(M); Z) −→ H`(BDiff∂(M#Wk); Z)
is an isomorphism if 2` ≤ rk(M)− 3 and an epimorphism when 2` ≤ rk(M)− 1.
1.4. Methodology
In this section we give an overview of the methods used in the proof of
Theorem 1.3.2. To prove our homological stability theorem, for each integer k ≥ 2
we construct a highly connected, semi-simplicial space X•(M)k, which admits an
action of the topological group Diff∂(M) that is transitive on the zero-simplicies.
The semi-simplicial space X•(M)k is defined roughly as follows. Let W ′k denote the
manifold with boundary obtained from Wk by removing an open disk. The space of
p-simplices, Xp(M)k, is defined to be the space of ordered (p+ 1)-tuples (φ0, . . . , φp)
where φi : W
′
k −→ M is an embedding for i = 0, . . . , p and φj(W ′k) ∩ φl(W ′k) = ∅
if j 6= l. With this definition, a p-simplex in Xp(M)k can be viewed as a particular
way of splitting a connected-sum factor of W
#(p+1)
k off of M .
The majority of the technical work of this paper is devoted to proving
that if M is 2-connected and k is odd, then the geometric realization |X•(M)k|
is 1
2
(rk(M) − 4)-connected. This is established in Section VIII and uses all of
the techniques developed throughout the rest of the thesis. In Section IX it is
13
shown how high connectivity of |X•(M)k| implies Theorem 1.3.2. This uses a
spectral sequence an argument essentially identical to (9, Section 5.2) and thus, the
majority of the technical work of this paper is devoted to proving that |X•(M)k| is
1
2
(rk(M)− 4)-connected.
Remark 1.4.1. This method of proving a homological stability theorem by
construction of a highly connected semi-simplicial space (or simplicial complex)
analogous to X•(M)k, is a fairly standard method and has been used to prove
homological stability theorems in many other contexts. A general overview of this
method of proving homological stability is given in (39) and many examples are
treated there as well.
The semi-simplicial space X•(M)k is very difficult to study directly. In order
to prove that its geometric realization is 1
2
(rk(M)− 4)-connected, we must compare
it to an auxiliary simplicial complex L(piτ2n(M))k whose definition is based on an
algebraic structure and thus can be analyzed via combinatorial methods. A p-
simplex in the simplicial complex L(piτ2n(M))k is defined to be a set of (p+ 1)-many,
pairwise orthogonal morphisms of linking forms (piτ2n(W
′
k), b) −→ (piτ2n(M), b),
which mimic the pairwise disjoint embeddings W ′k → M from the semi-simplicial
space X•(M)k. In Section 4.3, we prove that the geometric realization |L(piτ2n(M))k|
is 1
2
(rk(M) − 4)-connected (see Theorem 4.3.2) using the simplicial techniques
developed in Chapter II and the algebraic properties of bilinear forms defined on
finite groups. This result is similar to the classical homological stability theorem of
R. Charney from (7).
To compare |X•(M)k| to L(piτ2n(M))k, we consider a map |X•(M)k| −→
|L(piτ2n(M))k| induced by sending an embedding ϕ : W ′k −→ M , which
represents a 0-simplex in X•(M)k, to its induced morphism of linking forms,
14
ϕ∗ : (piτ2n(W
′
k), b) −→ (piτ2n(M), b), which represents a vertex in L(piτ2n(M))k.
With the high-connectivity of L(piτ2n(M))k established, to prove that |X•(M)k| is
highly connected it will suffice to prove that the above map induces an injection
on homotopy groups pij( ) when j ≤ 12(rk(M) − 4). This requires several new
geometric constructions which are of independent interest. In particular, we need
a technique for realizing morphisms (piτ2n(W
′
k), b) −→ (piτ2n(M), b) by actual
embeddings W ′k →M .
To solve the problem of realizing morphisms of linking forms by actual
embeddings of manifolds, we will need a suitable geometric model for the linking
form based on Z/k-manifolds and their intersections. Roughly, an m-dimensional
Z/k-manifold is a manifold with singularities whose local structure at the
singularity is of the form Rn−1 × C(〈k〉), where 〈k〉 denotes the set of k distinct
points and C(〈k〉) is the cone formed over that set. Z/k-manifolds can be used
to represent cycles in homology groups with Z/k-coefficients and thus they arise
for us in our analysis of the linking form. Sections V through VII, as well as the
appendices, are devoted to developing the intersection theory of Z/k-manifolds.
Our Z/k-manifold methods are largely inspired by the work of Morgan and
Sullivan in (33). However, this thesis contains several (to our knowledge) new
results regarding Z/k-manifolds, and other types of manifolds with singularities,
that don’t currently exist in the literature. These new results/techniques include:
– An h-principle for immersions of Z/k-manifolds into a smooth (non-singular)
manifold (Appendix 11.2).
– A technique for eliminating the self-intersections of an immersion of a Z/k-
manifold (Appendix 11.4).
15
– A method for modifying the intersections of embedded Z/k-manifolds when
a certain bordism invariant vanishes (Appendix ??). In the case that the
intersection in question is a zero dimensional manifold, this technique can
be viewed as a Z/k-version of the Whitney-trick (24, Theorem 6.6) used in
the proof of the h-cobordism theorem.
These techniques enable us to get a handle geometrically on the linking form and
they are all necessary to analyze the map which compares X•(M)k and L(M)k.
The main place where these singularity techniques are employed is in the proof of
Lemma 8.1.2.
Remark 1.4.2. Our main homological stability result requires the integer k to be
odd. The source of this restriction on the integer k is the technical result Theorem
10.5.1 and Theorem 11.6.1. If these theorems could be upgraded to include the case
that k is even, then Theorem 1.3.2 could be upgraded to include the case where k is
even as well.
It is also desirable to have a result analogous Theorem 1.3.1 for manifolds of
dimension 4n + 3. The key technical result in this paper for which the condition
that our manifolds be (4n + 1)-dimensional is required is Theorem 5.4.1 and
Corollary 11.5.3 (which is used to prove Theorem 5.4.1). If a version of Theorem
5.4.1 were to be extended to apply to manifolds of dimension 4n + 3, then an
analogue of the main result of this paper could be obtained for (4n+ 3)-dimensional
manifolds. However, the diffeomorphism classification of highly-connected manifolds
of dimension 4n + 3 (see (37)) is more involved than the classification in the
dimension 4n + 1 case, and so some other difficulties beyond Theorem 5.4.1 arise
as well.
16
We will treat both of these extraordinary cases, where k is even or the
manifolds are of dimension 4n+ 3, in a sequel to this paper.
1.5. Organization
Chapter II is a recollection of some basic definitions and results about
simplicial complexes and semi-simplicial spaces. In Chapter III we describe the
classification of (2n − 1)-connected, (2n + 1)-dimensional manifolds in terms of
linking forms. In Chapter IV we define the primary semi-simplicial space X•(M)k.
Then define the simplicial complex of linking forms L(piτ2n(M)k), and prove that it
is highly-connected. In Chapters V, VI, and VII we give the necessary background
on Z/k-manifolds used in the proof of Theorem 1.3.2. In these three sections we
state all of the necessary technical results regarding the intersections of immersions
and embeddings of Z/k-manifolds, but we put off most of the difficult proofs until
Appendix X and XI. In Chapter VIII we prove that the geometric realization
|X•(M)k| is highly connected. In Chapter IX we show how high-connectivity of
|X•(M)k| implies Theorem 1.3.2. In Appendix X and Appendix XI, we prove
several technical results regarding the intersections of immersions and embeddings
of Z/k-manifolds that were used earlier in the paper.
17
CHAPTER II
SIMPLICIAL TECHNIQUES
In this section we recall a number of simplicial techniques that we will need to use
throughout the paper. We will need to consider a variety of different simplicial
complexes and semi-simplicial spaces.
2.1. Cohen-Macaulay Complexes
Let X be a simplicial complex. Recall that the link of a simplex σ < X,
denoted by lkX(σ), is defined to be the subcomplex of X consisting of all simplices
ζ disjoint from σ, for which there exists a simplex ξ such that both σ and ζ are
faces of ξ.
We now present a key definition that will be used throughout the paper.
Definition 2.1.1. A simplicial complex X is said to be weakly Cohen-Macaulay of
dimension n if it is (n − 1)-connected and the link of any p-simplex is (n − p − 2)-
connected. In this case we write ωCM(X) ≥ n. The complex X is said to be locally
weakly Cohen-Macaulay of dimension n if the link of any simplex is (n − p − 2)-
connected (but no global connectivity is required on X itself). In this case we shall
write lCM(X) ≥ n.
We have some basic results about the links of simplices and Cohen-Macaulay
complexes. The following two lemmas are taken directly from (11, Lemmas 2.1 and
2.3).
Lemma 2.1.1. If ωCM(X) ≥ n and σ < X is a p-simplex, then ωCM(lkX(σ)) ≥
n− p− 1.
18
Definition 2.1.2. We say that a simplicial map f : X −→ Y is simplexwise
injective if its restriction to each simplex of X is injective, i.e. if v, v′ are adjacent
vertices in X, then h(v) 6= h(v′).
Lemma 2.1.2. Let f : X −→ Y be a simplicial map of simplicial complexes. Then
the following conditions are equivalent.
i. f is simplexwise injective,
ii. f(lkX(σ)) ⊂ lkY (f(v)) for all vertices v ∈ X,
iii. f(lkX(v)) ⊂ lkY (f(v)) for all vertices v ∈ X,
iv. the image of any 1-simplex in X is a (non-degeneate) 1-simplex in Y .
The next theorem is proven in (11, Section 2.1) and is a generalization of the
“Coloring Lemma” of Hatcher and Wahl from (15, Lemma 3.1). We include the
proof here since the result is so fundamental to us.
Theorem 2.1.3. Let X be a simplicial complex with lCM(X) ≥ n, let f : ∂In →
|X| be a map which is simplicial with respect to some PL triangulation of ∂In, and
h : In → |X| be a null-homotopy of f . Then the triangulation of ∂In extends
to a PL triangulation of In, and h is homotopic relative ∂In, to a simplicial map
g : In → |X| with the property that g(lkIn(v)) ≤ lkX(g(v)) for any interior vertex
v ∈ Int(In). In particular, g is simplexwise injective on the interior of In.
Proof. Since |X| is in particular (n − 1)-connected, we may extend f to a map
h : In −→ |X| which, by the simplicial approximation theorem, may be assumed
simplicial with respect to some PL triangulation of In extending the given one on
∂In.
19
Now, let us say that a simplex σ ∈ In is bad if every vertex v ∈ σ is contained
in a 1-simplex {v, v′} ⊂ σ with h(v) = h(v′). We will describe a procedure which
replaces the simplicial map h : In −→ X, by changing both h and the triangulation
of In, to a new map g : In −→ X with no bad simplices in the interior of In.
If all bad simplices are contained in ∂In, then we are done. If not, let σ < In
be a bad simplex not contained in ∂In, of maximal dimension p. By the definition
of bad simplex, the integer p must be greater than zero. Furthermore, we must
have h(lkIn(σ)) ⊂ lkX(h(σ)), since otherwise we could join a simplex in lkIn(σ) to
σ and get a new bad simplex of larger dimension. Now |σ| ⊂ In, so h restricts to a
map
∂In−p ∼= lkIn(σ) −→ lkX(h(σ)).
By badness of σ, the image h(σ) must be a simplex of dimension less than or
equal to p − 1, since otherwise h|σ would be injective. Since ωCM(X) ≥ n and
dim(h(σ)) ≤ p− 1, it follows that
ωCM(lkX(h(σ)) ≥ n− (p− 1)− 1 = n− p,
and in particular lkX(h(σ)) is (n − p − 1)-connected, so h|lkIn (σ) extends to a PL
map
In−p ∼= C(lkIn(σ)) hˆ−→ lkX(h(σ))
where C(lkIn(σ)) denotes he cone over lkIn(σ). By induction on n, we may assume
that hˆ is simplicial with respect to a PL triangulation of C(lkIn(σ)) which extends
the triangulation of lkIn(σ), and such that all bad simplices of hˆ are in ∂I
n−p =
20
lk(σ). We may extend this map by forming the join with h|∂σ to get a map
σ ? lkIn(σ) ∼= (∂σ) ? (C(lkIn(σ)) hˆ−→ X,
which we may finally extend to In by setting it equal to h outside of σ ? lkIn(σ) ⊂
In. The new map hˆ has fewer bad simplices of dimension p. By continuing
this process we eliminate all interior bad simplices in finitely many steps. This
completes the proof of the theorem.
Next we prove a result (Corollary 2.1.4) which will be employed several times
in Section VIII. This result, along with the property defined in Definition 2.1.3,
abstracts and isolates the key technique used in the proof of (11, Lemma 5.4).
Definition 2.1.3. Let f : X −→ Y be a simplicial map. The map f is said to have
the link lifting property if for any vertex y ∈ Y , the following condition holds: given
any subcomplex K ≤ X with f(K) ≤ lkY (y), there exists a vertex x ∈ X with
f(x) = y such that K ≤ lkX(x).
Corollary 2.1.4. Let X and Y be simplicial complexes and let f : X −→ Y be a
simplicial map. Suppose that the following conditions are met:
i. f has the link lifting property,
ii. lCM(Y ) ≥ n.
Then the induced map |f |∗ : pij(|X|) −→ pij(|Y |) is injective for all j ≤ n − 1.
Furthermore, suppose that in addition to properties i. and ii., f satisfies
iii. f(lkX(ζ)) ≤ lkY (f(ζ)) for all simplices ζ < X.
Then it follows that lCM(X) ≥ n.
21
Proof. For l + 1 ≤ n, let h : ∂I l+1 −→ |X| be a map which is simplicial with respect
to some PL triangulation of ∂I l+1, and let H : I l+1 −→ |Y | be a null-homotopy of
the composition |f | ◦ h, i.e. H|∂Il+1 = |f | ◦ h. To prove that |f |∗ : pil(|X|) −→ pil(|Y |)
is injective for all l ≤ n− 1, it will suffice to construct a lift Ĥ of H that makes the
diagram
∂I l+1 _

h // |X|
|f |

I l+1 H //
Ĥ
77
|Y |
commute. Since lCM(Y ) ≥ n, by Theorem 2.1.3 there exists a PL triangulation
of I l+1 that extends the chosen PL triangulation on ∂I l+1. Furthermore, we may
arrange that the map H satisfy H(lkIl(x)) ≤ lkY (H(x)) for any interior vertex
x ∈ Int(I l+1) (without altering the original definition of H on the boundary ∂I l+1).
We construct the lift Ĥ by inductively choosing lifts of each vertex in Int(I l+1) as
follows.
Suppose that Ĥ has already been defined on a full subcomplex K ≤ I l+1 (we
may assume that ∂I l+1 ≤ K). Let v ∈ I l+1 be a vertex in the compliment of K.
Let 〈K, v〉 denote the full subcomplex of I l+1 generated by the vertices of K and
v. We will use the link lifting property of f to extend the domain of Ĥ to 〈K, v〉.
Consider the subcomplex K ′ := K ∩ lkIl(v). We have H(K ′) ≤ lkY (H(v)) (recall
that by applying Theorem 2.1.3, we arranged for H to have this property in the
above paragraph). By the link lifting property of f , we may then choose a vertex
vˆ ∈ Y with f(vˆ) = H(v), such that Ĥ(K ′) ≤ lkX(vˆ). We then define Ĥ(v) = vˆ.
The fact that Ĥ(K ′) ≤ lkX(vˆ), implies that the definition Ĥ(v) = vˆ determines a
well defined simplicial map from 〈K, v〉, that extends the definition of Ĥ on K. By
repeating this process, we can extend the lift Ĥ over all of I l+1 inductively. This
22
establishes the existence of the lift Ĥ. It follows that |f |∗ : pil(|X|) −→ pil(|Y |) is
injective for all l < n.
Assume now that in addition to properties i. and ii. we have f(lkX(σ)) ≤
lkY (f(σ)) for all simplices σ < X. We will show that lCM(X) ≥ n. Let ζ ≤ X be a
p-simplex. Since f has the link lifting property, it follows that the map
f |lkX(ζ) : lkX(ζ) −→ lkY (f(ζ)) (2.1)
obtained by restricting f has the link lifting property as well. Since lCM(Y ) ≥ n,
it follows from (9, Lemma 2.2) that lCM [lkY (f(ζ))] ≥ n − p − 1. It follows from
the result proven in the previous paragraph that the map induced by (2.1) on pij( )
is injective for j ≤ n − p − 2. Since lkY (f(ζ)) is (n − p − 2)-connected, it follows
that lkX(ζ) is (n − p − 2)-connected as well. This proves that lCM(X) ≥ n and
completes the proof of the result.
Remark 2.1.1. The main technical challenge in this paper will be to prove that a
certain simplicial map (see (8.2) and Section 8.1) has the link lifting property. This
is established in the proof of Lemma 8.1.2 but it uses the geometric techniques
regarding Z/k-manifolds developed throughout Sections V, VI, VII and in the
appendix.
We will also need the useful following proposition, which was proven in (11,
Section 2.1). We will use it in the proof of Theorem 4.3.2.
Proposition 2.1.5. Let X be a simplicial complex and let Y ⊂ X be a full
subcomplex. Let n be an integer with the property that for each p-simplex σ < X,
the complex Y ∩ lkX(σ) is (n − p − 1)-connected. Then the inclusion |Y | ↪→ |X| is
n-connected.
23
2.2. Topological Flag Complexes
We will need to work with a certain class of semi-simplicial spaces called
topological flag complexes (see (? , Definition 6.1)).
Definition 2.2.1. Let X• be a semi-simplicial space. We say that X• is a
topological flag complex if for each integer p ≥ 0,
i. the map Xp −→ (X0)×(p+1) to the (p + 1)-fold product (which takes a p-
simplex to its (p + 1) vertices) is a homeomorphism onto its image, which is
an open subset,
ii. a tuple (v0, . . . , vp) ∈ (X0)×(p+1) lies in the image of Xp if and only if (vi, vj) ∈
X1 for all i < j.
If X• is a topological flag complex, we may denote any p-simplex x ∈ Xp by a
(p+ 1)-tuple (x0, . . . , xp) of zero-simplices.
Definition 2.2.2. Let X• be a topological flag complex and let x = (x0, . . . , xp) ∈
Xp be a p-simplex. The link of x, denoted by X•(x) ⊂ X•, is defined to be the
sub-semi-simplicial space whose l-simplices are given by the space of all ordered
lists (y0, . . . , yl) ∈ Xl such that the list (x0, . . . , xp, y0, . . . , y`) ∈ (X0)×(p+`+2), is a
(p+ `+ 1)-simplex.
It is easily verified that the link X•(x) is a topological flag complex as well.
The topological flag complex X• is said to be weakly Cohen-Macaulay of dimension
n if its geometric realization is (n − 1)-connected and if for any p-simplex x ∈ Xp,
the geometric realization of the link |X•(x)| is (n− p− 2)-connected. In this case we
write ωCM(X•) ≥ n.
24
The main result from this section is a result about the discretization of a
topological flag complex.
Definition 2.2.3. Let X• be a semi-simplicial space. Let Xδ• be the semi-simplicial
set defined by setting Xδp equal to the discrete topological space with underlying
set equal to Xp, for each integer p ≥ 0. We will call the semi-simplicial set Xδ• the
discretization of X•.
The following theorem is proven by repackaging several results from (11). In
particular, the proof is basically the same as the proof of (11, Theorem 5.5). We
provide a sketch of the proof here and we provide references to the key technical
lemmas employed from (11).
Theorem 2.2.1. Let X• be a topological flag complex and suppose that
ωCM(Xδ•) ≥ n. Then the geometric realization |X•| is (n− 1)-connected.
Proof Sketch. For integers p, q ≥ 0, let Yp,q = Xp+q+1 be toplogized as a subspace
of the product (X0)
×p × (Xδ0)×q. The assignment [p, q] 7→ Yp,q defines a bi-semi-
simplicial space with augmentations
ε : Y•,• −→ X•, δ : Y•,• −→ Xδ• .
This doubly augmented bi-semi-simplicial space is analogous to the one considered
in (11, Definition 5.6). Let ι : Xδ• −→ X• be the map induced by the identity. By
(11, Lemma 5.7), there exists a homotopy of maps,
|ι| ◦ |δ| ' || : |Y•,•| −→ |X•|. (2.2)
25
For each integer p, consider the map
|Yp,•| −→ Xp (2.3)
induced by . By how Y•,• was constructed, it follows from (11, Proposition 2.8)
that for each p, (2.3) is a Serre-microfibration. For any x ∈ Xp, the fibre over x is
equal to the space |Xδ•(x)|, where Xδ•(x) is the link of the p-simplex x, as defined
in Definition 2.2.2. Since ωCM(Xδ•) ≥ n, this implies that the fibre of (2.3) over
any x ∈ Xp is (n − p − 2)-connected. Using the fact that this map is a Serre-
microfibration, (11, Proposition 2.6) then implies that (2.3) is (n− p− 1)-connected.
It then follows by (11, Proposition 2.7) that the map
|| : |Y•,•| −→ |X•| (2.4)
is (n − 1)-connected. The homotopy from (2.2) implies that the map |ι| : |Xδ• | −→
|X•| induces a surjection on homotopy groups pij( ) for all j ≤ n − 1. The proof of
the theorem then follows from the fact that |Xδ• | is (n− 1)-connected by hypothesis.
2.3. Transitive Group Actions
In order to prove our homological stability theorem, we will need to consider
groups acting on simplicial spaces and simplicial complexes. We will need a
technique for determining when such actions are transitive. For the lemma that
follows, let X• be a topological flag complex, let G be a topological group, and let
G×X• −→ X•, (g, σ) 7→ g · σ
26
be a continuous group action.
Lemma 2.3.1. Let G and X• be as above and suppose that the following conditions
hold:
– for any 1-simplex (v, w) ∈ X1, there exists g ∈ G such that g · v = w,
– for any two vertices x, y that lie on the same path-component of X0, there
exists g ∈ G such that g · x = y,
– the geometric realization |X•| is path-connected.
Then for any two vertices x, y ∈ X0, there exists g ∈ G such that g · x = y.
Proof. We define an equivalence relation on the elements of X0 by setting x ∼ y
if there exists g ∈ G such that g · x = y. Since G is a group (and thus every
element has a multiplicative inverse), it follows that this relation is indeed an
equivalence relation, i.e. it is transitive, reflexive, and symmetric. By transitivity
of the relation, it follows from the in the statement of the lemma that x ∼ y if there
exists some zig-zag of edges connecting x and y. It also follows that x ∼ y if x and
y lie on the same path component of X0.
Let v, w ∈ X0 be any two zero simplices. We will prove that there exists
g ∈ G such that g · v = w. Since the geometric realization |X•| is path-connected,
it follows that there exists a vertex v′ in the path component containing v and a
vertex w′ in the path component containing w, such that v′ and w′ are connected
by a zig-zag of edges. We have v ∼ v′ ∼ w′ ∼ w, and thus v ∼ w. This concludes
the proof of the lemma.
There is a similar result for simplicial complexes which is proven in essentially
the same way as the previous lemma. We state the proposition without proof.
27
Proposition 2.3.2. Let K be a simplicial complex and let G×K −→ K be a group
action. Suppose |K| is path-connected and that for any edge {v, w} ≤ K, there
exists g ∈ G such that g · x = y. Then for any two vertices x, y ∈ K, there exists
g ∈ G such that g · x = y.
28
CHAPTER III
LINKING FORMS AND ODD-DIMENSIONAL MANIFOLDS
3.1. Linking Forms
The basic algebraic structure that we will encounter is that of a bilinear form
on a finite abelian group. For  = ±1, a pair (M, b) is said to be a (-symmetric)
linking form if M is a finite abelian group and b : M ⊗ M −→ Q/Z is an -
symmetric bilinear map. A morphism between linking forms is defined to be a
group homomorphism f : M −→ N such that bM(x, y) = bN(f(x), f(y)) for
all x, y ∈ M. We denote by L the category of all -symmetric linking forms. By
forming direct sums, L obtains the structure of an additive category.
Notational Convention 3.1.1. We will usually denote linking forms by their
underlying abelian group. We will always denote the bilinear map by b. If more
than one linking form is present, we will decorate b with a subscript so as to
eliminate ambiguity.
For M a linking form and N ≤ M a subgroup, N automatically inherits the
structure of a sub-linking form of M by restricting bM to N. We will denote
by N⊥ ≤ M the orthogonal compliment to N in M. Two sub-linking forms
N1,N2 ≤ M are said to be orthogonal if N1 ≤ N⊥2 , N2 ≤ N⊥1 , and N1 ∩N2 = 0.
If N1,N2 ≤ M are orthogonal sub-linking forms, we let N1 ⊥ N2 ≤ M denote the
sub-linking form given by the sum N1 + N2. If M1 and M2 are two linking forms,
the (external) direct sum M1 ⊕M2 obtains the structure of a linking form in a
29
natural way by setting
bM1⊕M2(x1 + x2, y1 + y2) = bM1(x1, y1) + bM1(x2, y2) for x1, y1 ∈M1, x2, y2 ∈M2.
(3.1)
We will always assume that the direct sum M1 ⊕M2 is equipped with the linking
form structure given by (3.1). An element M ∈ Ob(L) is said to be non-singular if
the duality homomorphism
T : M −→ HomAb(M,Q/Z), x 7→ b(x, ) (3.2)
is an isomorphism of abelian groups.
We will mainly need to consider the category L in the case where  = −1.
We denote by Ls−1 the full subcategory of L−1 consisting of linking forms that are
strictly skew symmetric, or in other words Ls−1 is the category of all linking forms
M for which bM(x, x) = 0 for all x ∈ M (even in the case when x is an element of
order 2).
We proceed to define certain basic, non-singular elements of Ls−1 as follows.
Definition 3.1.1. For a positive integer k ≥ 2, let Wk denote the abelian group
Z/k⊕Z/k. Let ρ and σ denote the standard generators (1, 0) and (0, 1) respectively.
We then let b : Wk −→ Q/Z be the −1-symmetric bilinear form determined by the
values
b(ρ, σ) = −b(σ, ρ) = 1
k
, b(ρ, ρ) = b(σ, σ) = 0. (3.3)
With b defined in this way, it follows that Wk is a non-singular object of Ls−1. It
follows easily that if k and ` are relatively prime, then Wk ⊕W` and Wk·` are
30
isomorphic as objects of Ls−1. For g ≥ 2 an integer, we will let Wgk denote the g-
fold direct sum (Wk)
⊕g.
For k ∈ N, let Ck denote the cyclic subgroup of Q/Z generated by the element
1/k mod 1. Any group homomorphism h : Wk −→ Q/Z must factor through the
inclusion Ck ↪→ Q/Z. Hence, it follows that the duality map from (3.2) induces an
isomorphism of abelian groups,
Wk
∼=−→ HomAb(Wk, Ck). (3.4)
Lemma 3.1.1. Let k ≥ 2 be a positive integer and let M ∈ Ob(Ls−1). Then any
morphism
f : Wk −→M
is split injective and there is an orthogonal direct sum decomposition, f(Wk) ⊥
f(Wk)
⊥ = M.
Proof. Let x and y denote the elements of M given by f(ρ) and f(σ) respectively
where ρ and σ are the standard generators of Wk. Let T : M −→ Hom(M,Q/Z)
denote the duality map from (3.2). Since both x and y have order k, it follows that
the homomorphisms
b(x, ), b(y, ) : M −→ Q/Z
factor through the inclusion Ck ↪→ Q/Z. Define a group homomorphism (which is
not a morphism of linking forms) by the formula
ϕ : M −→Wk, ϕ(z) = b(x, z) · ρ+ b(y, z) · σ.
31
It is clear that the kernel of ϕ is the orthogonal compliment f(Wk)
⊥ and that the
morphism f : Wk −→M gives a section of ϕ. This completes the proof.
The following theorem is a specialization of the classification theorem of Wall from
(38, Lemma 7). The classification of objects of Ls−1 is analogous to the classification
of finite abelian groups.
Theorem 3.1.2. Let M ∈ Ob(Ls−1) be non-singular. Then there is an isomorphism,
M ∼= W`1
p
n1
1
⊕ · · · ⊕W`r
pnrr
where pj is a prime number and `j and nj are positive integers for j = 1, . . . , r.
Furthermore, the above direct sum decomposition is unique up to isomorphism.
3.2. The Homological Linking Form
For what follows, let M be a manifold of dimension 2s + 1. Let Hτs (M ;Z) ≤
Hs(M ;Z) denote the torsion subgroup of Hs(M ;Z). Following (37), the homological
linking form b˜ : Hτs (M ;Z) ⊗ Hτs (M ;Z) −→ Q/Z is defined as follows. Let x, y ∈
τHs(M ;Z) and suppose that x has order r > 1. Represent x by a chain ξ and let
∂ζ = r · ξ. Then if y is represented by the chain χ, we define
b˜(x, y) = 1
r
[ζ ∩ χ] mod 1, (3.5)
where ζ ∩ χ denotes the algebraic intersection number associated to the two chains
(after being deformed so as to meet transversally). It is proven in (37, Page 274)
that b˜ is (−1)s+1-symmetric. We refer the reader to (37) for further details on this
construction.
32
Let piτs (M) ≤ pis(M) denote the torsion component of the homotopy group
pis(M). Using the homological linking form and the Hurewicz homomorphism h :
pis(M) −→ Hs(M), we can define a similar bilinear pairing
b : piτs (M)⊗ piτs (M) −→ Q/Z; b(x, y) = b˜(h(x), h(y)). (3.6)
The pair (piτs (M), b) is a (−1)s+1-symmetric linking form in the sense of Section
3.1 and we will refer to it as the homotopical linking form associated to M . In the
case that M is (s− 1)-connected, the homotopical linking form is isomorphic to the
homological linking form by the Hurewicz theorem.
3.3. The Classification Theorem
We are mainly interested in manifolds which are (4n + 1)-dimensional with
n ≥ 2. In this case the homological (and homotopical) linking form is anti-
symmetric. It follows from this that b(x, x) = 0 whenever x is of odd order. The
following lemma of Wall from (37) implies that for (4n + 1)-dimensional manifolds
for n ≥ 2, the linking form is strictly skew symmetric.
Lemma 3.3.1. For n ≥ 2, let M be a (2n − 1)-connected, (4n + 1)-dimensional
manifold. Then b(x, x) = 0 for all x ∈ piτ2n(M).
It follows from Lemma 3.3.1 that if M is a (2n − 1)-connected, (4n + 1)-
dimensional manifold (i.e. M ∈ W4n+1), then the homotopical linking form
(piτ2n(M), b) is an object of the category Ls−. If M is closed (or has boundary a
homotopy sphere), then (piτ2n(M), b) is non-singular. The following theorem is a
specialization of Wall’s classification theorem (37, Theorem 7).
33
Theorem 3.3.2. For n ≥ 2, two manifolds M1,M2 ∈ W¯S4n+1 are almost
diffeomorphic if and only if:
i. There is an isomorphism of Q-vector spaces, pi2n(M1)⊗Q ∼= pi2n(M2)⊗Q.
ii. There is an isomorphism of linking forms, (piτ2n(M1), b)
∼= (piτ2n(M2), b).
Furthermore, given any Q-vector space V and non-singular linking form M ∈ Ls−,
there exists an element M ∈ W¯S4n+1 such that, pi2n(M)⊗Q ∼= V and (piτ2n(M), b) ∼=
(M, bM).
Using the above classification theorem and the classification of skew
symmetric linking forms from Theorem 3.1.2, we may specify certain basic
manifolds.
Definition 3.3.1. For each integer k ≥ 2, fix a manifold Wk ∈ W¯S4n+1 which
satisfies:
(a) the homotopical linking form associated to Wk is isomorphic to Wk,
(b) pi2n(Wk)⊗Q = 0.
It follows from Theorem 3.3.2 that every element of W¯S4n+1 is almost
diffeomorphic (i.e. diffeomorphic up to connect-sum with a homotopy sphere) to
the connected sum of copies of Wk and copies of S
2n×S2n+1. The manifolds Wk are
the subject of our main result, Theorem 1.3.2.
Remark 3.3.1. The closed, stably parallelizable manifolds Wk ∈ W¯S4n+1 are
uniquely determined by conditions (a) and (b) up to almost diffeomorphism. For
each k, let W ′k denote the manifold obtained from Wk by removing an open disk. It
34
follows from (37, Theorem 7) that W ′k is determined by conditions (a) and (b) up
to diffeomorphism.
35
CHAPTER IV
THE PRIMARY COMPLEXES
4.1. The Complex of Connected-Sum Decompositions
Fix integers k, n ≥ 2. Let Wk denote the closed (4n+ 1)-dimensional manifold
Wk defined in Section 3.3. We will make a slight alteration of Wk as follows. Let
W ′k denote the manifold obtained from Wk by removing an open disk. Choose an
oriented embedding
α : {1} ×D4n −→ ∂W ′k.
We then define W¯k to be the manifold obtained by attaching [0, 1] × D4n to W ′k by
the embedding α, i.e.
W¯k := ([0, 1]×D4n) ∪αW ′k. (4.1)
Let M be a (4n + 1)-dimensional manifold with non-empty boundary. Fix an
embedding
a : [0,∞)× R4n −→M
with a−1(∂M) = {0} × R4n.
Definition 4.1.1. Let M and a : [0,∞)× R4n −→ M be as above and let k ≥ 2 be
an integer. We define a semi-simplicial space X•(M,a)k as follows:
(i) Let X0(M,a)k be the set of pairs (φ, t), where t ∈ R and φ : W¯k → M is an
embedding that satisfies the following condition: there exists  > 0 such that
for (s, z) ∈ [0, ) ×D4n ⊂ W¯k, the equality φ(s, z) = a(s, z + te1) is satisfied
(e1 ∈ R4n denotes the first basis vector).
36
(ii) For an integer p ≥ 0, Xp(M,a)k is defined to be the set of ordered (p + 1)-
tuples
((φ0, t0), . . . , (φp, tp)) ∈ (X0(M,a)k)×(p+1)
such that t0 < · · · < tp and φi(W¯k) ∩ φj(W¯k) = ∅ whenever i 6= j.
iii. For each p, the space Xp(M,a)k is topologized in the C
∞-topology as a
subspace of the product (Emb(W¯k,M)× R)×(p+1).
iv. The assignment [p] 7→ Xp(M,a)k makes X•(M,a)k into a semi-simplicial space
where the i-th face map Xp(M,a)k → Xp−1(M,a)k is given by
((φ0, t0, . . . , (φp, tp)) 7→ ((φ0, t0, . . . , (̂φi, ti), . . . , (φp, tp)).
It is easy to verify that X•(M,a)k is a topological flag complex. For any
0-simplex (φ, t) ∈ X0(M,a)k, it follows from condition i. that the number t is
determined by the embedding φ. For this reason we will usually drop the number t
when denoting elements of X0(M,a)k.
We now state a consequence of connectivity of the geometric realization
|X•(M,a)k|, using Lemma 2.3.1. This is essentially the same as (9, Proposition
4.4) and so we only give a sketch of the proof.
Proposition 4.1.1 (Transitivity). For n ≥ 2, let M be a (4n + 1)-dimensional
manifold with non-empty boundary. Let k ≥ 2 be an integer, and let φ0 and φ1 be
elements of X0(M,a)k. Suppose that the geometric realization |X•(M,a)k| is path
connected. Then there exists a diffeomorphism ψ : M
∼=−→ M , isotopic to the
identity when restricted to the boundary, such that ψ ◦ φ0 = φ1.
37
Proof Sketch. Let a : [0,∞)×R4n −→M be the embedding used in the definition of
X•(M,a). Let Diff(M,a) denote the group of diffeomorphisms ψ : M −→M with
ψ(a([0,∞)× R4n)) ⊂ a([0,∞)× R4n)
and such that ψ|∂M is isotopic to the identity. This group acts on X•(M,a)k, and
by Lemma 2.3.1 it will suffice to show that for (φ0, φ1) ∈ X1(M,a)k, there exists
ψ ∈ Diff(M,a) such that ψ ◦ φ0 = φ1. Let U ⊂ M be a collar neighborhood of the
boundary of M , that contains a([0,∞) × R4n). The union φ0(W¯k) ∪ φ1(W¯k) ∪ U is
diffeomorphic to manifold
Wk#(∂M × [0, 1])#Wk. (4.2)
To find the desired diffeomorphism ψ, it will suffice to construct a diffeomorphism
of (4.2), that is isotopic to the identity on the first boundary component, is equal
to the identity on the second boundary component, and that permutes the two
embedded copies of W ′k, that come from the two connected-sum factors. Such
a diffeomorphism can be constructed “by hand” using the same procedure that
was employed in the proof of (9, Proposition 4.4). We leave the details of this
construction to the reader.
The next proposition is proven in the same way as (9, Corollary 4.5), using
Proposition 4.2.1.
Proposition 4.1.2 (Cancelation). Let M and N be (4n+ 1)-dimensional manifolds
with non-empty boundaries, equipped with a specified identification, ∂M = ∂N .
For k ≥ 2, suppose that there exists a diffeomorphism M#Wk
∼=−→ N#Wk, equal
38
to the identity when restricted to the boundary. Then if |X•(M#Wk, a)k| is path-
connected, there exists a diffeomorphism M
∼=−→ N which is equal to the identity
when restricted to the boundary.
The main theorem that we will need is the following. Recall from (1.9) the
k-rank rk(M), of a (4n+ 1)-dimensional manifold M .
Theorem 4.1.3. Let n, k ≥ 2 be integers with k odd. Let M be a 2-connected,
(4n + 1)-dimensional manifold with non-empty boundary. Let g ∈ N be an integer
such that rk(M) ≥ g. Then the geometric realization |X•(M,a)k| is 12(g − 4)-
connected.
The majority of the technical work of thesis is devoted to proving the
above theorem. The proof will finally be given in Section VIII using techniques
developed throughout the rest of the paper. Once this theorem is proven, the proof
of the main theorem of this thesis, Theorem 1.3.2, follows by a spectral sequence
argument given in Section IX.
4.2. The Complex of Linking Forms
The topological flag complex X•(M,a)k introduced in the previous section is
very difficult to study directly. Indeed, for each integer p ≥ 0, the space Xp(M,a)k
is a space of codimension-zero embeddings, and relatively little is known about
the homotopy type of such spaces of embeddings in general. In order to prove
Theorem 4.1.3, we will need to compare X•(M,a)k to as simplicial complex, whose
definition is more algebraic than X•(M,a)k, and thus can be analyzed by discrete
and combinatorial methods. Below we define a simplicial complex, analogous to
the one from (11, Definition 3.1), that is based on the linking forms introduced in
Section 3.1.
39
Definition 4.2.1. Let M ∈ Ob(L−1) be a linking form and let k ≥ 2 be a positive
integer. We define L(M)k to be the simplicial complex whose vertices are given by
morphisms f : Wk −→ M of linking forms. The set {f0, . . . , fp} is a p-simplex if
the sub-linking forms fi(Wk) ≤M are pairwise orthogonal.
Suppose that σ = {f0, . . . , fp} is a p-simplex in L(M)k. Let M′ ≤ M
denote the sub-linking-form given by the orthogonal compliment [
∑
fi(Wk)]
⊥. It
follows from the definition of the link of a simplex that there is an isomorphism of
simplicial complexes,
lkL(M)k(σ)
∼= L(M′)k. (4.3)
Below are two formal consequences of path connectivity of L(M)k. They are proven
in the exact same way as (11, Propositions 3.3 and 3.4).
Proposition 4.2.1 (Transitivity). If |L(M)k| is path-connected and f0, f1 : Wk →
M are morphisms of linking forms, then there is an automorphism of linking forms
h : M→M such that f1 = h ◦ f0.
Proof. The group of linking form automorphisms Aut(M) acts on L(M)k by post-
composition with morphisms f : Wk −→ M. In order to prove the proposition, by
Proposition 2.3.2 it will suffice to show that given two morphisms f0, f1 : Wk →M
such that f0(Wk) and f1(Wk) are orthogonal, there exists ϕ ∈ Aut(M) such that
ϕ ◦ f0 = f1. So, let M′ ≤ M denote [f0(Wk) ⊥ f1(Wk)]⊥. By Proposition 3.1.1
there is an orthogonal splitting M = f0(Wk) ⊥ M′. We then define ϕ to be the
morphism determined by the data:
ϕ(fi(σ)) = f1−i(σ), ϕ(fi(ρ)) = fi−1(ρ), ϕ(v) = v for all v ∈M′,
where i = 0, 1. Clearly ϕ has the desired property. This concludes the proof.
40
Proposition 4.2.2 (Cancellation). Suppose that M and N are linking forms and
there is an isomorphism M ⊕Wk ∼= N ⊕Wk. If |L(M ⊕Wk)k| is path-connected,
then there is also an isomorphism M ∼= N.
4.3. High-Connectivity of the Linking Complex
In this section we prove a connectivity result for the complex L(M)k
introduced in the previous section. We will need to defined a notion of rank
analogous to (1.9) for skew-symmetric linking forms.
Definition 4.3.1. Let M be a linking form and let k ≥ 2 be a positive
integer. We define the k-rank of M to be the quantity, rk(M) = max{g ∈
N | there exists a morphism Wgk →M}. We then define the stable k-rank of M
to be the quantity, r¯k(M) = max{rk(M⊕Wgk)− g | g ∈ N.}
Corollary 4.3.1. Let f : Wgk −→ M be a morphism of linking forms. Then
r¯k(f(Wk)
⊥) ≥ r¯k(M)− g.
Proof. This follows immediately from the orthogonal splitting f(Wgk) ⊥ f(Wgk)⊥ =
M and the definition of the stable k-rank.
The main result that we will prove about the above complex is the following
theorem. The proof is very similar to the proof of (11, Theorem 3.2).
Theorem 4.3.2. Let g, k ∈ N and let M ∈ Ob(Ls−) be a linking form with r¯k(M) ≥
g. Then the geometric realization |L(M)k| is 12(g−4)-connected and lCM(L(M)k) ≥
1
2
(g − 1).
The proof of Theorem 4.3.2 follows the same inductive argument as the proof
of (11, Theorem 3.2). We will need two key algebraic results (Proposition 4.3.3
41
and Corollary 4.3.4) given below which are analogous to (11, Proposition 4.1 and
Corollary 4.2).
Proposition 4.3.3. Let k, g ∈ N with k ≥ 2. Let Aut(Wg+1k ) act on Wg+1k , and
consider the orbits of elements of Wk ⊕ 0 ≤ Wg+1k . We then have Aut(Wg+1k ) ·
(Wk ⊕ 0) = Wg+1k .
Proof. We will prove that for any v ∈ Wg+1k , there is an automorphism ϕ :
Wg+1k −→ Wg+1k such that v ∈ ϕ(Wk ⊕ 0). An element v ∈ Wg+1k is said
to be primitive if the subgroup 〈v〉 ≤ Wg+1k generated by v, splits as a direct
summand. Every element of Wg+1k is the integer multiple (reduced mod k) of a
primitive element. Hence it will suffice to prove the statement in the case that v is
a primitive element.
So, let v ∈ Wg+1k be a primitive element. Since the linking form Wg+1k is
non-singular and v is primitive, it follows that there exists w ∈ Wg+1k such that
b(w, v) = 1
k
mod 1. We may then define a morphism f : Wk −→ Wg+1k by
setting f(σ) = v and f(ρ) = w, where σ and ρ are the standard generators of Wk.
Consider the orthogonal splitting f(Wk) ⊥ f(Wk)⊥ = Wg+1k . Since both Wg+1k
and f(Wk) are non-singular, it follows that the orthogonal compliment f(Wk)
⊥ is
nonsingular as well. It then follows from the classification theorem (Theorem 3.1.2)
that there exists an isomorphism h : Wgk
∼=−→ f(Wk)⊥ (according to Theorem 3.1.2,
there is only one such way, up to isomorphism, to write Wg+1k as the direct sum
of Wk with another non-singular linking form). The morphism given by the direct
sum of maps
ϕ := f ⊕ h : Wk ⊕Wg+1k −→ f(W) ⊥ f(W)⊥,
42
is an isomorphism such that v ∈ ϕ(Wk ⊕ 0). This concludes the proof of the
proposition.
Corollary 4.3.4. Let M be a linking form with rk(M) ≥ g and let ϕ : M −→ Ck
be a group homomorphism. Then rk(Ker(ϕ)) ≥ g − 1. Similarly if r¯k(M) ≥ g then
r¯k(Ker(ϕ)) ≥ g − 1.
Proof. Since rk(M) ≥ g, there is a morphism f : Wgk −→ M. Consider the group
homomorphism given by
ϕ ◦ f : Wgk −→ Ck.
Since Wgk is non-singular, there exists v ∈ Wgk such that ϕ ◦ f(x) = b(v, x) for all
x ∈Wgk. By Proposition 4.3.3, there exists an automorphism h : Wgk −→Wgk such
that h−1(v) is in the sub-module Wk ⊕ 0 ≤ Wgk. It follows that the submodule
0⊕Wg−1k is contained in the kernel of the homomorphism given by the composition,
Wgk
h //Wgk
f //M
ϕ // Ck.
This implies that f(h(0 ⊕ Wg−1k )) is contained in the kernel of ϕ and thus
rk(Ker(ϕ)) ≥ g − 1.
Now suppose that r¯k(M) ≥ g and let ϕ : M −→ Ck be given. It follows
that rk(M ⊕Wjk) ≥ g for some integer j ≥ 0. Consider the map ϕ¯ given by the
composition,
M⊕Wjk
projM //M
ϕ // Ck.
By the result proven in the first paragraph, rk(Ker(ϕ¯)) ≥ g − 1. Clearly we have
Ker(ϕ¯) = Ker(ϕ)⊕Wjk. It then follows that r¯k(Ker(ϕ)) ≥ g− 1. This completes the
proof of the corollary.
43
The next proposition yields the first non-trivial case of Theorem 4.3.2.
Compare with (11, Proposition 4.3)
Proposition 4.3.5. If r¯k(M) ≥ 2, then L(M)k 6= ∅. If r¯k(M) ≥ 4, then L(M)k is
connected.
Proof. Let us first make the slightly stronger assumption that rk(M) ≥ 4. It follows
that there exists some morphism f0 : Wk −→ M such that rk(f0(Wk)⊥) ≥ 3.
Given any morphism f : Wk −→ M, we have a homomorphism of abelian groups
f0(Wk)
⊥ −→ M −→ f(Wk), where the first map is the inclusion and the second
is orthogonal projection. The kernel of this map is the intersection f0(Wk)
⊥ ∩
f(Wk)
⊥. Since Wk = Z/k ⊕ Z/k ∼= Ck ⊕ Ck (as an abelian group), it follows from
Corollary 4.3.4 that rk(f0(Wk)
⊥ ∩ f(Wk)⊥) ≥ 1. Thus, we can find a morphism
f ′ : Wk −→ f0(Wk)⊥ ∩ f(Wk)⊥.
It follows that the sets {f0, f} and {f0, f ′} are both 1-simplices, and so there is a
path of length 2 from f to f ′.
Now suppose that r¯k(M) ≥ 4. We then have an isomorphism of linking forms
M ⊕Wjk ∼= N ⊕Wjk for some j where rk(N) ≥ 4. By the first paragraph, L(N ⊕
Wjk)k is connected for all j ≥ 0, and so we may apply Proposition 4.2.2 inductively
to deduce that M ∼= N and thus rk(M) ≥ 4. We then apply the result of the first
paragraph to conclude that L(M)k is connected.
If r¯k(M) ≥ 2 we may write M⊕Wjk ∼= N⊕Wjk for some integer j and linking
form N such that rk(N) ≥ 2. We may then inductively apply Proposition 4.2.2 to
obtain an isomorphism f : M ⊕Wk
∼=−→ N ⊕Wk. The linking form M is then
isomorphic to the kernel of the orthogonal projection, N ⊕Wk −→ f(0 ⊕Wk).
44
Since rk(N ⊕Wk) ≥ 3 and Wk ∼= Ck ⊕ Ck, it follows from Corollary 4.3.4 that
rk(M) ≥ 1. From this, it follows that L(M)k is non-empty. This concludes the
proof of the proposition.
Proof of Theorem 4.3.2. We proceed by induction on g. The base case of the
induction, which is the case of the theorem where g = 4 and r¯(M) ≥ 4, follows
immediately from Proposition 4.3.5. Now suppose that the theorem holds for the
g − 1 case. Let M be a linking form with r¯k(M) ≥ g and g ≥ 4. By Proposition
4.3.5 there exists a morphism f : Wk −→ M and by Corollary 4.3.1 it follows
that r¯k(f(Wk)
⊥) ≥ g − 1. Let M′ denote the orthogonal compliment f(Wk)⊥
and consider the subgroup M′ ⊥ 〈f(σ)〉 ≤ M, where σ is one of the standard
generators of Wk (M
′ ⊥ 〈f(σ)〉 indicates an orthogonal direct sum). The chain
of inclusions M′ ↪→ M′ ⊥ 〈f(σ)〉 ↪→ M induces a chain of embeddings of sub-
simplicial-complexes
L(M′)k
i1 // L(M′ ⊥ 〈f(σ)〉)k i2 // L(M)k. (4.4)
The composition is null-homotopic since the vertex in L(M)k determined by the
morphism f : Wk −→M, is adjacent to every simplex in the subcomplex L(M′)k ≤
L(M)k. To prove that L(M)k is
1
2
(g − 4)-connected, we apply Proposition 2.1.5 to
the maps i1 and i2 with n :=
1
2
(g − 4). Since L(M′)k is (n − 1)-connected by the
induction assumption (recall that r¯(M′) ≥ g − 1), Proposition 2.1.5 together with
the fact that i2 ◦ i1 is null-homotopic will imply that L(M)k is 12(g − 4)-connected.
Let ξ be a p-simplex of L(M′ ⊥ 〈f(σ)〉)k. The linking form on the subgroup
f(σ) ≤ M′ is trivial and thus it follows that the projection homomorphism, pi :
M′ ⊥ 〈f(σ)〉 −→M′ preserves the linking form structure. Thus, there is an induced
45
simplicial map
p¯i : L(M′ ⊥ 〈f(σ)〉)k −→ L(M′)k,
and it follows easily that i1 is a section of p¯i. It follows from (4.3) that there is an
equality of simplicial complexes,
[lkL(M′⊥〈f(σ)〉)k(ξ)] ∩ L(M′)k = lkL(M′)k(p¯i(ξ)).
Since r¯k(M
′) ≥ g − 1, the induction assumption (which is that lCM(L(M′)k) ≥
1
2
(g − 2)) implies that the above complex is
1
2
(g − 2)− p− 2 = (n− p− 1)− connected,
where recall, n = 1
2
(g − 4). Proposition 2.1.5 then implies that the map i1 is n-
connected.
We now focus on the map i2. Since b(σ, σ) = 0, it follows that the subgroup
M′ ⊥ 〈f(σ)〉 ≤M
is precisely the orthogonal compliment of 〈f(σ)〉 in M. Let ζ := {f0, . . . , fp} ≤
L(M)k be a p-simplex, and denote M
′′ := [
∑
(fi(Wk))]
⊥ ≤M. We have,
L(M′ ⊥ 〈f(σ)〉)k ∩ lkL(M)(ζ) = L(M′′ ∩ 〈f(σ)〉⊥)k. (4.5)
Corollary 4.3.1 implies that r¯k(M
′′) ≥ g − p − 1. Passing to the kernel of the
homomorphism
b( , f(σ))|M′′ : M′′ −→ Ck,
46
reduces the stable k-rank by 1, and so we have r¯k(M
′′ ∩ 〈f(σ)〉⊥) ≥ g − p − 2. By
the induction assumption, it follows that the complex L(M′′ ∩ 〈f(σ)〉⊥)k is at least
1
2
(g − p− 2− 4) ≥ (n− p− 1)− connected.
By Proposition 2.1.5 it follows that the inclusion i2 is n-connected. Combining with
the previous paragraph implies that i2 ◦ i1 is n-connected. It then follows that
L(M)k is n =
1
2
(g − 4)-connected since i2 ◦ i1 is null-homotopic.
The fact that lCM(L(M)k) ≥ 12(g − 1) is proven as follows. Let ζ =
{f0, . . . , fp} ≤ L(M)k be a p-simplex and let V denote the orthogonal compliment
[
∑
fi(W)]
⊥. We have r¯k(V) ≥ g − p − 1. By (4.3) we have lkL(M)k(ζ) ∼= L(V)k
and so by the induction assumption it follows that | lkL(M)k(ζ)| is 12(g − p − 1 − 4)-
connected. The inequality
1
2
(g − p− 1− 4) = 1
2
(g − p− 1)− 2 ≥ 1
2
(g − 1)− p− 2
implies that | lkL(M)k(ζ)| is (12(g − 1) − p − 2)-connected. This proves that
lCM(L(M)k) ≥ 12(g − 1) and concludes the proof of the Theorem.
47
CHAPTER V
Z/K-MANIFOLDS
5.1. Basic Definitions
One of the main tools we will use to study the diffeomorphism groups of
odd dimensional manifolds will be manifolds with certain types of Baas-Sullivan
singularities, namely Z/k-manifolds (which in this paper we refer to as 〈k〉-
manifolds). We will use these manifolds to construct a geometric model for the
linking form. Here we give an overview of the definition and basic properties of
such manifolds. For further reference on Z/k-manifolds or manifolds with general
Baas-Sullivan singularities, see (3), (5), and (33).
Notational Convention 5.1.1. For a positive integer k, we let 〈k〉 denote the
set consisting of k-elements, {1, . . . , k}. We will consider this set to be a zero-
dimensional manifold.
Definition 5.1.1. Let k be a positive integer. Let P be a p-dimensional smooth
manifold equipped with the following extra structure:
i. The boundary of P has the decomposition, ∂P = ∂0P ∪ ∂1P where ∂0P and
∂1P are (p− 1)-dimensional manifolds with boundary and
∂0,1P := (∂0P ) ∩ (∂1P ) = ∂(∂0P ) = ∂(∂1P )
is a (d− 2)-dimensional closed manifold.
ii. There is a manifold βP and diffeomorphism Φ : ∂1P
∼=−→ βP × 〈k〉.
48
With P , βP, and Φ as above, the pair (P,Φ) is said to be a 〈k〉-manifold. The
diffeomorphism Φ is referred to as the structure-map and the manifold βP is called
the Bockstein.
Notational Convention 5.1.2. We will usually drop the structure-map from
the notation and denote P := (P,Φ). We will always denote the structure-map
associated to a 〈k〉-manifold by the same capital greek letter Φ. If another 〈k〉-
manifold is present, say Q, we will decorate the structure map with the subscript
Q, i.e. ΦQ.
Any smooth manifold M is automatically a 〈k〉-manifold by setting ∂0M =
∂M , ∂1M = ∅, and βM = ∅. Such a 〈k〉-manifold M with ∂1M = ∅, βM = ∅ is said
to be non-singular.
Now, let P be a 〈k〉-manifold as in the above definition. Notice that the
diffeomorphism Φ maps the submanifold ∂0,1P ⊂ ∂1P diffeomorphically onto ∂(βP ).
In this way, if we set
∂0(∂0P ) := ∅, ∂1(∂0P ) := (∂0P ) ∩ (∂1P ) = ∂0,1P, and β(∂0P ) = ∂(βP ),
the pair ∂0P := (∂0P, Φ|∂0,1P ) is a 〈k〉-manifold. We will refer to ∂0P as the
boundary of P . If ∂0P = ∅, then P is said to be a closed 〈k〉-manifold.
Given a 〈k〉-manifold P , one can construct a manifold with cone-type
singularities in a natural way as follows.
Definition 5.1.2. Let P be a 〈k〉- manifold. Let Φ¯ : ∂1P −→ βP be the map
given by the composition ∂1P
Φ
∼=
// βP × 〈k〉 projβP // βP. We define P̂ to
be the quotient space obtained from P by identifying points x, y ∈ ∂1P if and only
if Φ¯(x) = Φ¯(y).
49
We will need to consider maps from 〈k〉-manifolds to non-singular manifolds.
Definition 5.1.3. Let P be a 〈k〉-manifold and let X be a topological
space. A map f : P −→ X is said to be a 〈k〉-map if there exists a map
fβ : βP → X such that the restriction of f to ∂1P has the factorization
∂1P
Φ¯ // βP
fβ // X, where Φ¯ : ∂1P −→ βP is the map from
Definition 5.1.2. Clearly the map fβ is uniquely determined by f .
We denote by Maps〈k〉(P,X) the space of 〈k〉-maps P → M , topologized
as a subspace of Maps(P,X) with the compact-open topology. It is immediate
that any 〈k〉-map f : P → X induces a unique map f̂ : P̂ −→ X and that the
correspondence, f 7→ f̂ induces a homeomorphism, Maps〈k〉(P,X) ∼= Maps(P̂ , X).
Throughout the paper we will denote by fˆ : P̂ −→ Y , the map induced by the 〈k〉-
map f . In the case that X is a smooth manifold, f is said to be a smooth 〈k〉-map
if both f and fβ are both smooth.
5.2. Bordism of 〈k〉-Manifolds
We will need to consider the oriented bordism groups of 〈k〉-manifolds. For a
space X and non-negative integer j, we denote by ΩSOj (X)〈k〉 the bordism group of
j-dimensional, oriented 〈k〉-manifolds associated to X. We refer the reader to (5)
and (33) for precise details of the definitions. We have the following Theorem from
(5).
Theorem 5.2.1. For any space X and integer k ≥ 2, there is a long exact
sequence:
· · · // ΩSOj (X) ×k // ΩSOj (X)
jk // ΩSOj (X)〈k〉
β // ΩSOj−1(X) // · · ·
(5.1)
50
where ×k denotes multiplication by the integer k, jk is induced by inclusion (since
an oriented smooth manifold is an oriented 〈k〉-manifold), and β is the map induced
by P 7→ βP .
It is immediate from the above long exact sequence that for all integers k ≥ 2,
there are isomorphisms
ΩSO0 (pt.)〈k〉 ∼= Z/k and ΩSO1 (pt.)〈k〉 ∼= 0. (5.2)
5.3. Z/k-Homotopy Groups
For integers k, n ≥ 2, let M(Z/k, n) denote the n-th Z/k-Moore-space. Recall
that M(Z/k, n) is uniquely determined up to homotopy by the calculation,
Hj(M(Z/k, n); Z) ∼=

Z/k if j = n,
Z if j = 0,
0 else.
For a space X, we denote by pin(X; Z/k) the set of based homotopy classes of
maps M(Z/k, n) −→ X. Since M(Z/k, n) is a suspension when n ≥ 2, the set
pin(X; Z/k) has the structure of a group, which is abelian when n ≥ 3.
For integers n, k ≥ 2, we define a 〈k〉-manifold which will play the role of the
sphere in the category of 〈k〉-manifolds.
Construction 5.3.1. Choose an embedding Φ′ : Dn × 〈k〉 −→ Sn. Let V nk denote
the manifold obtained from Sn by removing the interior of Φ′(Dn × 〈k〉) from Sn.
The inverse of the restriction of the map Φ′ to ∂Dn × 〈k〉 induces a diffeomorphism,
51
Φ : ∂V nk
∼=−→ Sn−1 × 〈k〉. By setting βV nk = Sn−1, the above diffeomorphism Φ gives
V nk the structure of a closed 〈k〉-manifold.
Let V̂ nk denote the singular space obtained from V
n
k as in Definition 5.1.2. An
elementary calculation shows that,
Hj(V̂
n
k )
∼=

Z/k if j = n− 1 or 0,
Z⊕(k−1) if j = 1,
and pi1(V̂
n
k )
∼= Z?(k−1), (5.3)
where Z?(k−1) denotes the free group on (k − 1)-generators. It follows that the
Moore-space M(Z/k, n− 1) can be constructed from V̂ nk by attaching (k − 1)-many
2-cells, one for each generator of the fundamental group. This yields the following
result.
Lemma 5.3.1. Let X be a 2-connected space and let k ≥ 2 and n ≥ 3 be
integers. The inclusion map V̂ nk ↪→ M(Z/k, n − 1) induces a bijection of sets,
pi0(Maps〈k〉(V
n
k , X))
∼=−→ pin−1(X;Z/k).
Proof. Since X is simply connected, any map V̂ nk −→ X extends to a map
M(Z/k, n− 1) −→ X and since X is 2-connected, it follows that any such extension
is unique up to homotopy. This proves that the inclusion V̂ nk ↪→ M(Z/k, n − 1)
induces a bijection pi0(Maps(V̂
n
k , X))
∼= pin−1(X;Z/k). The lemma then follows
from composing this bijection with the natural bijection, pi0(Maps〈k〉(V
n
k , X))
∼=−→
pi0(Maps(V̂
n
k , X)).
Corollary 5.3.2. Let X be a 2-connected space and let k ≥ 2 and n ≥ 3 be
integers. Let x ∈ pin−1(X) be an element of order k. Then there exists a 〈k〉-map
f : V nk −→ X such that the associated map fβ : Sn−1 −→ X is a representative of x.
52
Proof. The cofibre sequence Sj
×k // Sj //M(Z/k, j) induces a long exact
sequence,
· · · // pin(X) ×k // pin(X) // pin−1(X;Z/k) ∂ // pin−1(X) ×k // pin−1(X) // · · ·
It follows that if x ∈ pin−1(X) is of order k, then there is an element y ∈
pin−1(X;Z/k) such that ∂y = x. Let rβ : pi0(Maps〈k〉(V nk , X)) −→ pin−1(X) denote
the map induced by, f 7→ fβ. It follows from the construction of the the map ∂ in
the above long exact sequence that the diagram,
pi0(Maps〈k〉(V
n
k , X))
rβ
**
∼= // pin−1(X;Z/k)
∂

pin−1(X)
commutes, where the upper horizontal map is the bijection from Lemma 5.3.1. The
result then follows from commutativity of this diagram.
5.4. Immersions and Embeddings of 〈k〉-Manifolds
We will need to consider immersions and embeddings of a 〈k〉-manifold into
a smooth manifold. For what follows, let P be a 〈k〉-manifold and let M be a
manifold.
Definition 5.4.1. A 〈k〉-map f : P −→ M is said to be a 〈k〉-immersion if it is
an immersion when considering P as a smooth manifold with boundary. Two 〈k〉-
immersions f, g : P −→ M are said to be regularly homotopic if there exists a
homotopy Ft : P −→ M with F0 = f and F1 = g such that Ft is a 〈k〉-immersion
for all t ∈ [0, 1].
53
In addition to immersions we will mainly need to deal with embeddings of
〈k〉-manifolds.
Definition 5.4.2. A 〈k〉-immersion f : P −→ M is said to be a 〈k〉-embedding if
the induced map f̂ : P̂ −→M is an embedding.
The main result about 〈k〉-embeddings that we will use is the following. The
proof is given in Section 11.6, using the techniques developed throughout all of
Section XI.
Theorem 5.4.1. Let n ≥ 2 be an integer and let k > 2 be an odd integer. Let
M be a 2-connected, oriented manifold of dimension 4n + 1. Then any 〈k〉-map
f : V 2n+1k −→M is homotopic through 〈k〉-maps to a 〈k〉-embedding.
The following corollary follows immediately by combining Theorem 5.4.1 with
Corollary 5.3.2.
Corollary 5.4.2. Let n ≥ 2 be an integer and let k > 2 be an odd integer. Let
M be a 2-connected, oriented manifold of dimension 4n + 1. Let x ∈ pi2n(M) be a
class of order k. Then there exists a 〈k〉-embedding f : V 2n+1k −→ M such that the
embedding fβ : βV
2n+1
k = S
2n −→M is a representative of the class x.
54
CHAPTER VI
〈K,L〉-MANIFOLDS
6.1. Basic definitions
We will have to consider certain spaces with more complicated singularity
structure than that of the 〈k〉-manifolds encountered in the previous section.
Definition 6.1.1. Let k and l be positive integers. Let N be a smooth d-
dimensional manifold equipped with the following extra structure:
i. The boundary ∂N has the decomposition,
∂N = ∂0N ∪ ∂1N ∪ ∂2N
such that ∂0N , ∂1N and ∂2N are (d − 1)-dimensional manifolds, the
intersections
∂0,1N := ∂0,1N, ∂0,2N := ∂0N ∩ ∂2N, ∂1,2N := ∂1N ∩ ∂2N
are (d− 2)-dimensional manifolds, and
∂0,1,2N := ∂0N ∩ ∂1N ∩ ∂2N
is a (d− 3)-dimensional closed manifold.
55
ii. There exist manifolds β1N, β2N, and β1,2N , and diffeomorphisms
Φ1 : ∂1N
∼= // β1N × 〈k〉,
Φ2 : ∂2N
∼= // β2N × 〈l〉,
Φ1,2 : ∂1,2N
∼= // β1,2N × 〈k〉 × 〈l〉,
such that the maps
Φ1 ◦ Φ−11,2 : β1,2N× 〈k〉× 〈l〉 // β1N×〈k〉,
Φ2 ◦ Φ−11,2 : β1,2N× 〈k〉× 〈l〉 // β1N×〈l〉,
are identical on the direct factors of 〈k〉 and 〈l〉 respectively.
With the above conditions satisfied, the 4-tuple N := (N,Φ1,Φ2,Φ1,2) is said to be
a 〈k, l〉-manifold of dimension d.
Remark 6.1.1. The above definition is a specialization of Σ-manifold from (5,
Definition 1.1.1) and a generalization of the definition of 〈k〉-manifold. In fact, any
〈k〉-manifold P is a 〈k, l〉-manifold with β2P = ∅.
As for the case with 〈k〉- manifolds, we will drop the structure maps
Φ1,Φ2,Φ1,2 from the notation and denote N := (N,Φ1,Φ2,Φ1,2). The manifold
∂0W is referred to as the boundary of the 〈k, l〉-manifold and is a 〈k, l〉-manifold in
its own right. A 〈k, l〉-manifold N is said to be closed if ∂0N = ∅.
From a 〈k, l〉-manifold N , one obtains a manifold with cone-type singularities
in the following way.
Definition 6.1.2. Let N be a 〈k, l〉-manifold. Let Φ¯1 : ∂1N −→ β1N be the
map defined by the composition ∂1N
Φ1
∼=
// β1N × 〈k〉
projβ1N // β1N. Define Φ¯2 :
56
∂2N −→ β2N similarly. We define N̂ to be the quotient space obtained from N by
identifying two points x, y if and only if for i = 1 or 2, both x and y are in ∂iW and
Φ¯i(x) = Φ¯i(y).
6.2. Oriented 〈k, l〉-Bordism
We will need to make use of the oriented bordism groups of 〈k, l〉-manifolds.
For any space X and non-negative integer j, we denote by ΩSOj (X)〈k,l〉 the j-th
〈k, l〉-bordism group associated to the space X. We refer the reader to (5) for
details on the definition. There are maps
β1 : Ω
SO
j (X)〈k,l〉 −→ ΩSOj−1(X)〈l〉, β2 : ΩSOj (X)〈k,l〉 −→ ΩSOj−1(X)〈k〉
defined by sending a 〈k, l〉-manifold N to β1N and β2N respectively. We also have
maps
j1 : Ω
SO
j (X)〈k〉 −→ ΩSOj (X)〈k,l〉, j2 : ΩSOj (X)〈l〉 −→ ΩSOj (X)〈k,l〉
defined by considering a 〈k〉-manifold or an 〈l〉-manifold as a 〈k, l〉-manifold. We
have the following theorem from (5).
Theorem 6.2.1. The following sequences are exact,
· · · // ΩSOj (X)〈l〉 ×l // ΩSOj (X)〈l〉
j1 // ΩSOj (X)〈k,l〉
β1 // ΩSOj−1(X)〈l〉 // · · ·
· · · // ΩSOj (X)〈k〉 ×k // ΩSOj (X)〈k〉
j2 // ΩSOj (X)〈k,l〉
β2 // ΩSOj−1(X)〈k〉 // · · ·
Using the isomorphisms ΩSO0 (pt.)〈k〉 ∼= Z/k and ΩSO1 (pt.)〈k〉 = 0, we obtain the
following basic calculations using the above exact sequence.
57
Corollary 6.2.2. For any two integers k, l ≥ 2 we have the following
isomorphisms,
ΩSO0 (pt.)〈k,l〉 ∼= Z/ gcd(k, l) and ΩSO1 (pt.)〈k,l〉 ∼= Z/ gcd(k, l).
In particular we have,
ΩSO0 (pt.)〈k,k〉 ∼= Z/k and ΩSO1 (pt.)〈k,k〉 ∼= Z/k.
6.3. 1-dimensional, Closed, Oriented, 〈k, k〉-Manifolds
We will need to consider 1-dimensional 〈k, k〉-manifolds. They will arise for us
as the intersections of (n + 1)-dimensional 〈k〉-manifolds immersed in a (2n + 1)-
dimensional manifold. Denote by Ak the space [0, 1]× 〈k〉. By setting
∂1Ak = {0} × 〈k〉 and ∂2Ak = {1} × 〈k〉,
Ak naturally has the structure of a closed 〈k, k〉-manifold with, β1Ak = 〈1〉 =
β2Ak (the single point space). We denote by +Ak the oriented 〈k, k〉-manifold with
orientation induced by the standard orientation on [0, 1]. We denote by −Ak the
〈k, k〉-manifold equipped with the opposite orientation. It follows that
β1(±Ak) = ±〈1〉 and β2(±Ak) = ∓〈1〉. (6.1)
Using the fact that the map βi : Ω
SO
1 (pt.)〈k,k〉 −→ ΩSO0 (pt.)〈k〉 for i = 1, 2 is an
isomorphism (this follows from Corollary 6.2.2 and the exact sequence in Theorem
6.2.1), we have the following proposition.
58
Proposition 6.3.1. The oriented, closed, 〈k, k〉 manifold +Ak represents a
generator for ΩSO1 (pt.)〈k,k〉. Furthermore, any oriented, closed, 1-dimensional 〈k, k〉-
manifold that represents a generator of ΩSO1 (pt.)〈k,k〉, is of the form
(+Ak × 〈r〉) unionsq (−Ak × 〈s〉) unionsqX,
where r, s ∈ N are such that r − s is relatively prime to k, and where X is some
null-bordant 〈k, k〉-manifold such that β1X = ∅ or β2X = ∅ (in other words, X has
the structure of 〈k〉-manifold).
Throughout, we will consider the element of ΩSO1 (pt.)〈k,k〉 determined by the
oriented 〈k, k〉-manifold +Ak to be the standard generator.
59
CHAPTER VII
INTERSECTION THEORY OF Z/K-MANIFOLDS
In this section and the next two sections after, we will discuss the
intersections of embeddings of 〈k〉-manifolds.
7.1. Preliminaries
Here we review some of the basics about intersections of embedded smooth
manifolds and introduce some terminology and notation.
For what follows, let M , X, and Y be oriented smooth manifolds of dimension
m, r, and s respectively and let t denote the integer r + s−m. Let
ϕ : (X, ∂X) −→ (M,∂M) and ψ : (Y, ∂Y ) −→ (M,∂M) (7.1)
be smooth, transversal maps such that ϕ(∂X) ∩ ψ(∂Y ) = ∅ (for these two maps
to be transversal, we mean that the product map ϕ × ψ : X × Y −→ M ×M is
transverse to the diagonal submanifold 4M ⊂ M ×M). We let ϕ t ψ denote the
transverse pull-back (ϕ × ψ)−1(4M), which is a closed submanifold of X × Y of
dimension t. The orientations on X, Y , and M induce an orientation on ϕ t ψ and
thus ϕ t ψ determines a bordism class in ΩSOt (pt.) which we denote by Λt(ϕ, ψ;M).
It follows easily that, Λt(ϕ, ψ;M) = (−1)(m−s)·(m−r)Λt(ψ, ϕ;M).
7.2. Intersections of 〈k〉-Manifolds
We now proceed to consider intersections of 〈k〉-manifolds. Let M be an
oriented manifold of dimension m, let X be an oriented manifold of dimension
60
r, and let P be an oriented 〈k〉-manifold of dimension p. Let t denote the integer
r + p−m. Let
ϕ : (X, ∂X) −→ (M,∂M) and f : (P, ∂0P ) −→ (M,∂M)
be a smooth map and a smooth 〈k〉-map respectively. Suppose that f and ϕ
are transversal and that f(∂0P ) ∩ ϕ(∂X) = ∅ (when we say that f and ϕ are
transversal, we mean that both f and fβ are transverse to ϕ as smooth maps). The
pull-back,
f t ϕ = (f × ϕ)−1(4M) ⊂ P ×X
has the structure of a closed 〈k〉-manifold as follows. We denote,
∂1(f t ϕ) := f |∂1P t ϕ and β(f t ϕ) := fβ t ϕ.
The factorization, ∂1P
Φ¯ // βP
fβ //M of the restriction map f |∂1P implies that
the diffeomorphism,
Φ× IdX : ∂1P ×X ∼= // (βP × 〈k〉)×X
maps ∂1(f t X) diffeomorphically onto β(f t X) × 〈k〉. It follows that f t ϕ has
the structure of a 〈k〉-manifold of dimension t = p + r − m. Furthermore, f t ϕ
inherits an orientation from the orientations of X, P and M .
Definition 7.2.1. Let f : (P, ∂0P ) −→ (M,∂M) and ϕ : (X, ∂X) −→ (M,∂M) be
exactly as above. We define Λtk(f, ϕ;M) ∈ ΩSOt (pt.)〈k〉 to be the oriented bordism
class determined by the pull-back f t ϕ and its induced orientation.
61
Recall from Section V the Bockstein homomorphism, β : ΩSOt (pt.)〈k〉 −→
ΩSOt−1(pt.). We have the following proposition.
Proposition 7.2.1. Let f : (P, ∂0P ) −→ (M,∂M) and ϕ : (X, ∂X) −→ (M,∂M) be
exactly as above. Then
β(Λtk(f, ϕ;M)) = Λ
t−1(fβ, ϕ;M),
where Λt−1(fβ, ϕ;M) ∈ ΩSOt−1(pt.) is the bordism class defined in Section 7.1.
7.3. 〈k, l〉-Manifolds and Intersections
We now consider the intersection of a 〈k〉-manifold with an 〈l〉-manifold. For
what follows, let P be an oriented 〈k〉-manifold of dimension p, let Q be an oriented
〈l〉-manifold of dimension q, and let M be an oriented manifold of dimension m.
Let
f : (P, ∂0P ) −→ (M,∂M) and g : (Q, ∂0Q) −→ (M,∂M)
be a smooth 〈k〉-map and a smooth 〈l〉-map respectively. Suppose that f and g
are transversal and that f(∂0P ) ∩ g(∂0Q) = ∅ (when we say that f and g are
transversal, we mean that f and fβ are each transverse to both g and gβ as smooth
maps). Let t denote the integer p + q − m. We will analyze the t-dimensional
submanifold
f t g = (f × g)−1(4M) ⊂ P ×Q.
62
The transversality condition on f and g implies that the space f t g, and the
subspaces
f |∂P t g ⊂ ∂P ×Q, f t g|∂Q ⊂ P × ∂Q, f |∂P t g|∂Q ⊂ ∂P × ∂Q,
fβ t g ⊂ βP ×Q, f t gβ ⊂ P × βQ, fβ t gβ ⊂ βP × βQ,
are all smooth submanifolds. We define
∂1(f t g) := f |∂P t g, ∂2(f t g) := f t g|∂Q, ∂1,2(f t g) := f |∂P t g|∂Q,
β1(f t g) := fβ t g, β2(f t g) := f t gβ, β1,2(f t g) := fβ t gβ.
The structure maps, ΦP : ∂P
∼=−→ βP × 〈k〉 and ΦQ : ∂Q
∼=−→ βQ × 〈l〉 induce
diffeomorphisms,
ΦP × Id : ∂P ×Q ∼= // βP × 〈k〉 ×Q,
Id× ΦQ : P × ∂Q ∼= // P × βQ× 〈l〉,
ΦP × ΦQ : ∂P × ∂Q ∼= // βP × 〈k〉 × βQ× 〈l〉.
(7.2)
The factorizations,
∂P
Φ¯P // βP
fβ //M,
∂Q
Φ¯Q // βQ
gβ //M,
of the restriction maps f |∂P and g|∂Q imply that the diffeomorphisms from (7.2)
map the submanifolds
∂1(f t g) ⊂ ∂P ×Q, ∂2(f t g) ⊂ P × ∂Q, and ∂1,2(f t g) ⊂ ∂P × ∂Q
diffeomorphically onto
β1(f t g)× 〈k〉, β2(f t g)× 〈l〉, and β1,2(f t g)× 〈k〉 × 〈l〉
63
respectively. It follows that f t g has the structure of an oriented 〈k, l〉-manifold of
dimension t = p+ q −m.
Definition 7.3.1. Let f : (P, ∂0P ) −→ (M,∂M) and g : (Q, ∂0Q) −→ (M,∂M)
be exactly as above. We denote by Λ1k,l(f, g;M) ∈ ΩSOt (pt.)〈k,l〉 the bordism class
determined by the pull-back f t g.
For the following proposition, recall from Section 5.2 the Bockstein
homomorphisms,
β1 : Ω
SO
t (pt.)〈k,l〉 −→ ΩSOt−1(pt.)〈l〉 and β2 : ΩSOt (pt.)〈k,l〉 −→ ΩSOt−1(pt.)〈k〉.
Proposition 7.3.1. The bordism class Λtk,l(f, g;M) ∈ ΩSOt (pt.)〈k,l〉 satisfies the
following equations
i. Λtk,l(f, g;M) = (−1)(m−p)·(m−q) · Λtl,k(g, f ;M),
ii. β1(Λ
t
k,l(f, g;M)) = Λ
t−1
l (fβ, g;M),
iii. β2(Λ
t
k,l(f, g;M)) = Λ
t−1
k (f, gβ;M).
7.4. Main Disjunction Theorem
We now discuss the main result that we will need to use regarding the
intersections of k-manifolds. We will need the following terminology.
Definition 7.4.1. Let M be a manifold. We will call a smooth, one parameter
family of diffeomorphisms Ψt : M −→ M with t ∈ [0, 1] and Ψ0 = IdM a diffeotopy.
For a subspace N ⊂ M , we say that Ψt is a diffeotopy relative N , and we write
Ψt : M −→M rel N , if in addition, Ψt|N = IdN for all t ∈ [0, 1].
64
The main case of intersections of 〈k〉 and 〈l〉-manifolds that we will need to
consider is the case when
k = l and dim(P ) + dim(Q)− dim(M) = 1.
For n ≥ 2, let M be an oriented manifold of dimension 4n + 1 and and let P and Q
be oriented k-manifolds of dimension 2n+ 1. Let
f : (P, ∂0P ) −→ (M,∂M) and g : (Q, ∂0Q) −→ (M,∂M) (7.3)
be transversal 〈k〉-embeddings such that f(∂0P ) ∩ g(∂0Q) = ∅. Suppose further that
M , P , and Q are 2-connected.
Theorem 7.4.1. With f and g the 〈k〉-embeddings given above, suppose that
Λ1k,k(f, g;M) = 0. If the integer k is odd, then there exists a diffeotopy
Ψt : M −→M rel ∂M
such that Ψ1(f(P )) ∩ g(Q) = ∅.
We also have:
Corollary 7.4.2. Suppose that the class Λ1k,k(f, g;M) ∈ ΩSO1 (pt.)〈k,k〉 is equal to the
class represented by the closed 1-dimensional 〈k, k〉-manifold +Ak. If k is odd, there
exists a diffeotopy Ψt : M −→ M rel ∂M such that the 〈k, k〉-manifold given by the
transverse pull-back (Ψ1 ◦ f) t g, is diffeomorphic to Ak.
Remark 7.4.1. Both of these results are proven in Section ?? (see Theorem 10.5.1
and Corollary 10.5.4). These above results are crucial in the proof of our main
65
homological stability theorem. The key place (only place) that they are used is
in the proof of Lemma 8.1.2.
7.5. Connection to the Linking Form
In practice we will need to consider intersections of 〈k〉 embeddings f, g :
V 2n+1k −→ M . We will need to relate Λ1k,k(f, g;M) to the homotopical linking form
b : piτ2n(M)⊗ piτ2n(M) −→ Q/Z. Let
Tk : Ω
SO
1 (pt.)〈k,k〉 −→ Q/Z
be the homomorphism given by the composition
ΩSO1 (pt.)〈k,k〉
+Ak 7→1 // Z/k 17→1/k // Q/Z.
The following proposition follows easily from the definition of the homological
linking form (3.5).
Proposition 7.5.1. Let M be a (4n+ 1)-dimensional, oriented manifold. Let
f, g : V 2n+1 −→M
be k-embeddings. Consider the homotopy classes [fβ], [gβ] ∈ piτ2n(M), which both
have order k. Then
b([fβ], [gβ]) = Tk(Λ
1
k,k(f, g;M)).
Combining this with Theorem 7.4.1 and Corollary 10.5.4 yields the following.
66
Corollary 7.5.2. Let
f, g : V 2n+1 −→M
be k-embeddings and suppose that b([fβ], [gβ]) = 0. Then there exists a diffeotopy
Ψt : M −→ M such that Ψ1(f(V 2n+1k ) ∩ g(V 2n+1k ) = ∅. Suppose now that
b([fβ], [gβ]) =
1
k
mod 1. Then there exists a diffeotopy Ψt : M −→ M such that
there is a diffeomorphism (Ψ1 ◦ f) t g ∼= +Ak.
67
CHAPTER VIII
HIGH CONNECTIVITY OF |X•(M,A)K |
Let n ≥ 2 be an integer. Let M be a 2-connected, (4n + 1)-dimensional
manifold with non-empty boundary. Fix an embedding a : [0,∞)× R4n −→M with
a−1(∂M) = {0} × R4n. Recall from Definition 4.1.1 the topological flag complex
X•(M,a)k. In this chapter we prove Theorem 4.1.3 which we restate again below
for the convenience of the reader. Recall from (1.9) the k-rank rk(M), of a (4n+ 1)-
dimensional manifold M .
Theorem 8.0.3. Let n, k ≥ 2 be integers with k odd. Let M be a 2-connected,
(4n + 1)-dimensional manifold with non-empty boundary. Let g ∈ N be an integer
such that rk(M) ≥ g. Then the geometric realization |X•(M,a)k| is 12(g − 4)-
connected.
The proof of this theorem will require several intermediate constructions.
8.1. The Complex of 〈k〉-Embeddings
Fix integers n, k ≥ 2. Let M be a manifold of dimension (4n + 1) with non-
empty boundary. Consider transversal 〈k〉-embeddings
ϕ0, ϕ1 : V 2n+1k −→M
such that the transverse pull-back ϕ0 t ϕ1 is diffeomorphic to Ak as a 〈k, k〉-
manifold. It follows that ϕ0(V 2n+1k ) ∩ ϕ1(V 2n+1k ) ∼= Âk, where Âk is the singular
space obtained from Ak as in Definition 6.1.2. It will be useful to have an abstract
model for the space given by the union, ϕ0(V 2n+1k ) ∪ ϕ1(V 2n+1k ).
68
Construction 8.1.1. To begin the construction, fix a point y ∈ Int(V 2n+1k ).
For i = 1, . . . , k, let ∂i1V
2n+1
k denote the component of the boundary given by
Φ−1(βV 2n+1k × {i}), where 〈k〉 = {1, . . . , k}. Let Φ¯ : ∂1V 2n+1k −→ βV 2n+1k be
the map used in Definition 5.1.2.
i. For i = 1, . . . , k, fix points xi ∈ ∂i1V 2n+1k such that, Φ¯(x1) = · · · = Φ¯(xk).
ii. For i = 1, . . . , k, choose embeddings γi : [0, 1] −→ V 2n+1k such that
γi(0) = xi, γ
−1
i (∂1V
2n+1
k ) = {0}, and γi(1) = y.
Then for each i, let γ¯i : [0, 1] −→ V 2n+1k be the embedding given by the
formula
γ¯i(t) = γ(1− t).
iii. Recall that Ak = [0, 1]× 〈k〉 = unionsqki=1[0, 1]. The maps
unionsqki=1γi : Ak −→ V 2n+1k and unionsqki=1 γ¯i : Ak −→ V 2n+1k ,
yield embeddings
Γ : Âk −→ V̂ 2n+1k and Γ¯ : Âk −→ V̂ 2n+1k .
iv. We define Y 2n+1k to be the space obtained by forming the push-out of the
diagram,
Âk
Γ
ww
Γ¯
''
V̂ 2n+1k V̂
2n+1
k
(8.1)
69
v. By applying the Mayer-Vietoris sequence and Van Kampen’s theorem we
compute,
Hs(Y
2n+1
k ;Z) ∼=

Z/k ⊕ Z/k if s = 2n,
Z⊕(k−1) if s = 1,
Z if s = 0,
pi1(Yk) ∼= Z?(k−1),
where Z?(k−1) denotes the free-group on (k − 1)-generators.
The next proposition follows easily by inspection.
Proposition 8.1.1. Let ϕ0, ϕ1 : V 2n+1k −→ M be transversal 〈k〉-embeddings
such that the pull-back is diffeomorphic to Ak as a 〈k, k〉-manifold. Then the union
ϕ0(V 2n+1k ) ∪ ϕ1(V 2n+1k ) is homeomorphic to the space Y 2n+1k .
Notation 8.1.1. Let ϕ = (ϕ0, ϕ1) be a pair of 〈k〉-embeddings ϕ0, ϕ1 : V 2n+1k −→
M such that the transverse pull-back is diffeomorphic to Ak as a 〈k, k〉-manifold.
We will denote by Yk(ϕ
0, ϕ1) the subspace of M given by the union ϕ0(V 2n+1k ) ∪
ϕ1(V 2n+1k ).
We now define a simplicial complex based on pairs of 〈k〉-embeddings,
V 2n+1k →M as above.
Definition 8.1.1. Let M and k be as above. Let K(M)k be the simplicial complex
with vertex set given by the set of all pairs (ϕ0, ϕ1) of transverse 〈k〉-embeddings
ϕ0, ϕ1 : V 2n+1k −→M
70
such that the transverse pull-back is diffeomorphic to Ak as a 〈k, k〉-manifold. A set
{(ϕ00, ϕ10) . . . , (ϕ0p, ϕ1p)}
of vertices forms a p-simplex if Yk(ϕ
0
i , ϕ
1
i ) ∩ Yk(ϕ0j , ϕ1j) = ∅ whenever i 6= j.
Now, recall from Section 4.2, the simplicial complex L(M)k associated to
an object M of the category L−1 of anti-symmetric linking forms. We will need
to compare the simplical complex K(M)k to the simplicial complex L(pi
τ
2n(M))k,
where (piτ2n(M), b) is the homotopical linking form associated to M , see (3.6). We
construct a simplicial map
F : K(M)k −→ L(piτ2n(M))k (8.2)
as follows. For a vertex ϕ = (ϕ0, ϕ1) ∈ K(M)k, let 〈[ϕ0β], [ϕ1β]〉 ≤ piτ2n(M) denote
the subgroup generated by the homotopy classes determined by the embeddings
ϕνβ : S
2n → M for ν = 0, 1. The classes [ϕνβ], ν = 0, 1 each have order k and
b([ϕ0β], [ϕ
1
β]) =
1
k
. It follows that the sub-linking form given by 〈[ϕ0β], [ϕ1β]〉 ≤ piτ2n(M)
is isomorphic to the standard non-singular linking form Wk from Definition 3.1.1.
The map F from (8.2), is then defined by sending a vertex ϕ to the morphism of
linking forms Wk → piτ2n(M) determined by
ρ 7→ [ϕ0β], σ 7→ [ϕ1β],
where ρ and σ are the standard generators of Wk. The disjointness condition from
condition ii. of Definition 8.1.1, implies that this formula preserves all adjacencies
and thus yields a well defined simplicial map. It follows easily that for any (4n+ 1)-
71
dimensional manifold M and integer k ≥ 2 that
rk(pi
τ
2n(M)) ≥ rk(M) (8.3)
where recall, rk(pi
τ
2n(M)) is the k-rank of the linking form (pi
τ
2n(M), b) as defined
in Definition 4.3.1, and rk(M) is the k-rank of the manifold M as defined in the
introduction.
Lemma 8.1.2. Let n, k ≥ 2 be integers with k odd. Let M be a 2-connected
manifold of dimension 4n + 1. Then the geometric realization |K(M)k| is
1
2
(rk(M)− 4)-connected and lCM(K(M)k) ≥ 12(rk(M)− 1).
Proof. Let rk(M) ≥ g. Since L(piτ2n(M))k is 12(g − 4)-connected and
lCM(L(piτ2n(M))k) ≥ 12(g − 1), the proof of the lemma will follow directly from
Corollary 2.1.4 once we verify two things:
i. the map F has the link lifting property (see Definition 2.1.3),
ii. F (lkK(M)k(ζ)) ≤ lkL(piτ2n(M))k(F (ζ)) for any simplex ζ ∈ K(M)k.
Property i. is proven by applying Corollary 5.4.2, Corollary 10.5.4, and Theorem
7.4.1 as follows. Let f : Wk −→ piτ2n(M) be a morphism of linking forms (which
determines a vertex in L(piτ2n(M))k). Let ρ, σ ∈Wk denote the standard generators
as defined in Section ??. The elements f(ρ), f(σ) ∈ piτ2n(M) have order k and thus
by Corollary 5.4.2 we may choose 〈k〉-embeddings ϕ0, ϕ1 : V 2n+1k −→ M such that
[ϕ0β] = f(ρ) and [ϕ
1
β] = f(σ). Furthermore, since b(f(ρ), f(σ)) =
1
k
mod 1, it
follows from Proposition 7.5.1 that,
Λ1k,k([ϕ
0], [ϕ1]) = [+Ak] ∈ ΩSO1 (pt.)〈k,k〉.
72
We then may apply Corollary 10.5.4 (or Corollary 7.4.2) so as to obtain an isotopy
of ϕ0 through 〈k〉-embeddings to a 〈k〉-embedding ϕ¯0, so that ϕ¯0 t ϕ1 ∼= Ak.
The pair (ϕ¯0, ϕ1) determines a vertex in K(M)k and clearly F ((ϕ¯
0, ϕ1)) = f . This
shows how to lift any vertex v ∈ L(piτ2n(M))k to a vertex vˆ ∈ K(M)k such that
F (vˆ) = v.
Now let f : Wk → piτ2n(M) be a morphism representing a vertex of
L(piτ2n(M))k, let f1, . . . , fm an arbitrary set of vertices in L(pi
τ
2n(M))k that are
adjacent to f , and let
(ϕ01, ϕ
1
1), . . . , (ϕ
0
m, ϕ
1
m)
be vertices in K(M)k with F ((ϕ
0
i , ϕ
1
i )) = fi, for i = 1, . . . ,m. To show that F
has the link lifting property, it will suffice to construct a vertex (ϕ¯0, ϕ¯1) ∈ K(M)k
mapping to f , such that (ϕ¯0, ϕ¯1) is adjacent to (ϕ0i , ϕ
1
i ) for i = 1, . . . ,m.
For each i and ν = 0, 1, let ϕνβ,i : S
2n = βV 2n+1k −→ M denote the
map associated to ϕνi and let [ϕ
ν
β,i] denote the associated class in pi2n(M). For
i = 1, . . . ,m we have:
b(f(ρ), [ϕνβ,i]) = b(f(σ), [ϕ
ν
β,i]) = 0 for ν = 0, 1. (8.4)
By Corollary 5.4.2, we may choose 〈k〉-embeddings ϕ0, ϕ1 : V 2n+1k −→ M with
[ϕ0β] = f(σ) and [ϕ
1
β] = f(ρ). By (8.4) we may inductively apply Corollary 7.5.2
(or Theorem 7.4.1) to find isotopies of ϕ0 and ϕ1 (through 〈k〉-embeddings) to new
〈k〉-embeddings
ϕ¯0, ϕ¯1 : V 2n+1k −→M
such that:
73
(a) Yk(ϕ
0
i , ϕ
1
i ) ∩ Yk(ϕ¯0, ϕ¯1) = ∅ for i = 1, . . .m,
(b) ϕ¯0 t ϕ¯1 ∼= Ak.
This proves that F has the link lifting property.
The fact that F (lkK(M)k(ζ)) ≤ lkL(piτ2n(M))k(F (ζ)) for any simplex ζ ∈ K(M)k
follows immediately from the fact that if φ, ψ : V 2n+1k −→ M are disjoint 〈k〉-
embeddings, then b([φβ], [ψβ]) = 0. This establishes property ii. and completes the
proof of the Lemma.
8.2. A Modification of K(M)k
Let (ϕ0, ϕ1) be a vertex of K(M)k and consider the subspace Yk(ϕ
0, ϕ1) ⊂ M .
We will need to make a further modification of Yk(ϕ
0, ϕ1) as follows.
Construction 8.2.1. Let (ϕ0, ϕ1) be as above. Since 2 < dim(M)/2, we may
choose an embedding
G : (unionsqk−1i=1D2i , unionsqk−1i=1 S1i ) // (M, Yk(ϕ0, ϕ1)) (8.5)
which satisfies the following conditions:
(a)
G(unionsqk−1i=1 Int(D2i ))
⋂
Yk(ϕ
0, ϕ1) = ∅.
(b) The maps
G|S1i : S1 −→ Yk(ϕ0, ϕ1) for i = 1, . . . , k − 1,
represent a minimal set of generators for pi1(Yk(ϕ
0, ϕ1)), which by Proposition
8.1.1 is the free group on k − 1 generators.
74
Given such an embedding G as in (8.5), we denote
Y Gk (ϕ
0, ϕ1) := Yk(ϕ
0, ϕ1)
⋃
G(unionsqk−1i=1D2i ). (8.6)
It follows from conditions i. and ii. above that Y Gk (ϕ
0, ϕ1) is simply connected and
that
Hs(Y
G
k (ϕ
0, ϕ1); Z) =

Z/k ⊕ Z/k if s = 2n,
Z if s = 0,
0 else.
It follows that Y Gk (ϕ
0, ϕ1) has the homotopy type of the Moore-space M(Z/k ⊕
Z/k, 2n) and hence is homotopy equivalent to the manifold W ′k. We will think of
Y Gk (ϕ
0, ϕ1) ↪→M as being a choice of embedding of the (2n+ 1)-skeleton of W ′k into
M .
Using the construction given above, we define a modification of the simplicial
complex K(M)k. Let M be a (4n + 1)-dimensional manifold with non-empty
boundary. Let
a : [0,∞)× R4n −→M
be an embedding with a−1(∂M) = {0} × R4n.
Definition 8.2.1. Let K¯(M,a)k be the simplicial complex whose vertices are given
by 4-tuples (ϕ,G, γ, t) which satisfy the following conditions:
i. ϕ = (ϕ0, ϕ1) is a vertex in K(M)k.
ii. G : (unionsqk−1i=1D2i , unionsqk−1i=1 S1i ) // (M, Yk(ϕ0, ϕ1)) is an embedding as in
Construction (8.2.1).
75
iii. t is a real number.
iv. γ : [0, 1] −→M is an embedded path which satisfies:
(a) γ−1(Y Gk (ϕ
0, ϕ1)) = {1},
(b) there exists  > 0 such that for s ∈ [0, ), the equality
γ(s) = a(s, te1) ∈ [0, 1]× R4n
is satisfied, where e1 ∈ R4n denotes the first basis vector.
A set of vertices {(ϕ0, G0, γ0, t0), . . . , (ϕp, Gp, γp, tp)} forms a p-simplex if and only if
(
γi([0, 1]) ∪ Y Gik (ϕ0i , ϕ1i )
)⋂(
γj([0, 1]) ∪ Y Gjk (ϕ0j , ϕ1j)
)
= ∅ whenever i 6= j.
There is a simplicial map
F¯ : K¯(M,a)k −→ K(M)k, (ϕ,G, γ, t) 7→ ϕ. (8.7)
Proposition 8.2.1. Let n, k ≥ 2 be integers with k odd. Let M be a 2-connected,
manifold of dimension 4n + 1 and let g ∈ N be such that rk(M) ≥ g. Then the
geometric realization |K¯(M)k| is 12(g − 4)-connected and lCM(K¯(M)k) ≥ 12(g − 1).
Proof. The proof of this proposition is proven by the same method as Lemma 8.1.2.
It is proven by verifying that the map F¯ from (8.7) has the link lifting property
(Definition 2.1.3) and that it preserves links. Since |K(M)k| is 12(g − 4)-connected
and lCM(K(M)k) ≥ 12(g − 1), we then may apply Corollary 2.1.4 to deduce the
claim of the proposition.
76
Let ϕ = (ϕ0, ϕ1) be a vertex in K(M)k. Let (ϕ1, G1, γ1, t1), . . . , (ϕm, Gm, γm, tm)
be vertices in K¯(M,a)k such that ϕi is adjacent to ϕ for i = 1, . . . ,m. Since
dim(M)/2 > 2, there is no obstruction to choosing an embedding G as in
Construction 8.2.1 (with respect to ϕ so as to construct Y Gk (ϕ
0, ϕ1)) so that the
image of G is disjoint from the images of Gi and γi for all i. Furthermore, with
G chosen, we may then choose an embedded path γ : [0, 1] −→ M , connecting
Y Gk (ϕ
0, ϕ1) to ∂M so as to yield a vertex (ϕ,G, γ, t) ∈ K¯(M,a)k, which maps to
ϕ under F¯ and is adjacent to (ϕi, Gi, γi, ti) for all i. This proves the fact that F¯
has the link lifting property. The fact that F¯ preserves links is immediate from the
definition of F¯ . This concludes the proof of the proposition.
8.3. Reconstructing Embeddings
Let (ϕ,G, γ, t) be a vertex in K¯(M,a)k. We will need to consider smooth
regular neighborhoods of the subspace Y Gk (ϕ
0, ϕ1) ∪ γ([0, 1]) ⊂ M. The following
lemma identifies the diffeomorphism type of such a regular neighborhood. Recall
from Section 4.1 the manifold W¯k = W
′
k ∪α ([0, 1] × D4n) used in the definition of
X•(M,a)k.
Lemma 8.3.1. Let (ϕ,G, γ, t) be a vertex in K¯(M,a)k. If k is odd then any closed
regular neighborhood U of the subspace Y Gk (ϕ
0, ϕ1)∪γ([0, 1]) ⊂ M , is diffeomorphic
to the manifold W¯k = W
′
k ∪α ([0, 1]×D4n).
Proof. By definition of regular neighborhood, the inclusion map Y Gk (ϕ
0, ϕ1) ↪→ U
is a homotopy equivalence (U collapses to Y Gk (ϕ
0, ϕ1), see (17)). The maps ϕ0β, ϕ
1
β :
S2n −→ U represent generators for pi2n(U) and since ϕ0 t ϕ1 ∼= Ak, it follows that
b([ϕ0β], [ϕ
1
β]) =
1
k
mod 1
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and hence, the linking form (piτ2n(U), b) is isomorphic to Wk. It follows from
Constructions 8.1.1 and 8.2.1 that the regular neighborhood U is (2n − 1)-
connected. Now, U is homotopy equivalent to the Moore-space M(Z/k ⊕ Z/k, 2n),
and so the set of isomorphism classes of (4n + 1)-dimensional vector bundles over
U is in bijective correspondence with the set [M(Z/k ⊕ Z/k, 2n), BSO]. Since
pi2n(BSO;Z/k) = 0 whenever k is odd, it follows that the tangent bundle TU → U
is trivial and thus U ∈ WS4n+1 (i.e. U is stably parallelizable). We will show
that the boundary ∂U is diffeomorphic to S4n. Once this is demonstrated, it will
follow from the classification theorem, Theorem 3.3.2 (and Remark 3.3.1), that U is
diffeomorphic to the manifold W ′k.
Since U is parallelizable, by (21, Theorem 5.1) it will be enough to show
that ∂U is homotopy equivalent to S4n. From Constructions 8.1.1, 8.2.1 and the
Universal Coefficient Theorem, we have
Hs(U ;Z) ∼=

Z/k ⊕ Z/k if s = 2n+ 1,
Z if s = 0,
0 else.
Using Lefschetz Duality it then follows that
Hs(U, ∂U ;Z) ∼=

Z/k ⊕ Z/k if s = 2n,
0 else.
Consider the long exact sequence on homology associated to (U, ∂U). It follows
immediately that ∂U is (2n−2)-connected and that the long exact sequence reduces
78
to
0 // H2n(∂U ;Z) // H2n(U ;Z) // H2n(U, ∂U ;Z) // H2n−1(∂U ;Z) // 0.
(8.8)
We claim that the map H2n(U,Z) → H2n(U, ∂U ;Z) is an isomorphism. To see this,
consider the commutative diagram
H2n(U ;Z) ∼=
x7→b(x, ) //

H2n(U ;Q/Z)
∼=

H2n(U, ∂U ;Z)
∼= // H2n+1(U ;Z).
In the above diagram the bottom-horizontal map is the Leftshetz duality
isomorphism, the right vertical map is the boundary homomorphism in the
Bockstein exact sequence (which in this case is an isomorphism), and the top-
horizontal map x 7→ b(x, ) is an isomorphism since the homological linking form
(H2n(U), b) is non-singular. It follows that the map H2n(U ;Z) → H2n(U, ∂U ;Z) is
indeed an isomorphism and it then follows from the exact sequence of (8.8) that ∂U
has the same homology type of S4n.
To prove that ∂U has the same homotopy type of S4n, we must show that
∂U is simply connected. To to this it will suffice to show that pii(U, ∂U) = 0 for
i = 1, 2. For i = 1, 2, let f : (Di, ∂Di) −→ (U, ∂U) be a map. Since
dim(U)− dim(Y Gk (ϕ0, ϕ1)) ≥ 3,
we may deform f so that its image is disjoint from Y Gk (ϕ
0, ϕ1). We then may then
find another (strictly smaller) regular neighborhood U ′ of Y Gk (ϕ
0, ϕ1) such that
79
U ′ ( U and f(Di) ⊂ U \ U ′. The class [f ] ∈ pii(U, ∂U) is in the image of the map
pii(U \ Int(U ′), ∂U) −→ pii(U, ∂U)
induced by inclusion. Using the uniqueness theorem for smooth regular
neighborhoods (see (17)), it follows that the manifold U \ Int(U ′) is an H-cobordism
from ∂U to ∂U ′ and so it follows thatpii(U \ Int(U ′), ∂U) = 0. This proves that
[f ] = 0 and thus pii(U, ∂U) = 0 since f was arbitrary. It follows by considering
the exact sequence on homotopy groups associated to the pair (U, ∂U) that ∂U is
simply connected.
Since ∂U is simply connected and has the homology type of a sphere, it
follows that ∂U is a homotopy sphere. It then follows from (21, Theorem 5.1) that
∂U is diffeomorphic to S4n since ∂U bounds a parallelizable manifold, namely U .
This concludes the proof of the lemma.
We now define a new simplicial complex.
Definition 8.3.1. Let K̂(M,a)k be the simplicial complex whose vertices are given
by triples (ϕ¯,Ψ, s) which satisfy the following conditions:
i. The 4-tuple ϕ¯ = (ϕ,G, γ, t) is a vertex in K¯(M,a)k.
ii. s is a real number.
ii. Ψ : W¯k × [s,∞) −→ M is a smooth family of embeddings W¯k ↪→ M that
satisfies the following:
(a) for each t ∈ [s,∞), the embedding Ψ( , t) : W¯k −→ M is an element of
X0(M,a)k,
80
(b) Y G(ϕ0, ϕ1) ∪ γ([0, 1]) ⊂ Ψ(W¯k, t) for all t ∈ [s,∞),
(c) for any neighborhood U of Y G(ϕ0, ϕ1)∪γ([0, 1]), there is tU ∈ [s,∞) such
that Ψ(W¯k, t) ⊂ U when t ≥ tU .
A set of vertices {(ϕ¯0,Ψ0, s0), . . . , (ϕ¯p,Ψp, sp)} forms a p-simplex if the associated
set {ϕ¯0, . . . , ϕ¯p} is a p-simplex in the complex K¯(M,a)k (no extra pairwise
condition on the Ψi and si are required).
By construction of K̂(M,a)k, there is a simplicial map,
F̂ : K̂(M,a)k −→ K¯(M,a)k, (ϕ¯,Ψ, s) 7→ ϕ¯. (8.9)
Proposition 8.3.2. Let n, k ≥ 2 be integers with k odd. Let M be a compact,
2-connected, manifold of dimension 4n + 1. Let g ∈ N be such that rk(M) ≥ g.
Then the geometric realization |K̂(M)k| is 12(g − 4)-connected and lCM(K¯(M)k) ≥
1
2
(g − 1).
Proof. The proof of this proposition again follows the same strategy as Lemma
8.1.2. We check that the map F̂ has the cone lifting property, that it preserves
links, and then we apply Corollary 2.1.4.
The fact that F̂ preserves links follows immediately from the definition. We
will verify the link lifting property. Let ϕ¯ = (ϕ,G, γ, t) be a vertex of K¯(M)k. Let
(ϕ¯1,Ψ1, s1), . . . , (ϕ¯m,Ψm, sm)
81
be a collection of vertices in K̂(M)k such that ϕ¯ is adjacent to ϕ¯i in K¯(M)k for
i = 1, . . . ,m. We will denote
Yk(ϕ¯) := Y
G
k (ϕ
0, ϕ1) ∪ γ([0, 1]). (8.10)
Let U ⊂ M be a regular neighborhood of Yk(ϕ¯). Since U collapses to Yk(ϕ¯) (by
definition of regular neighborhood), we may choose a one-parameter family of
embeddings:
ρ : U × [s,∞) −→ U (8.11)
which satisfies the following:
i. For all t ∈ [s,∞), the embedding ρt = ρ|U×{t} : U → U is the identity on
Yk(ϕ¯).
ii. Given any neighborhood U ′ ⊂ U of Yk(ϕ¯), there exists t′ > s such that
ρt(U) ⊂ U ′ for all t ≥ t′.
We call such an isotopy a compression isotopy of U to Yk(ϕ¯). By Lemma 8.3.1,
there exists a diffeomorphism Ψ : W¯k
∼=−→ U such that the composition W¯k Ψ−→
U ↪→ M satisfies the conditions of Definition 8.2.1. It then follows that the triple
(ϕ¯, Ψ ◦ ρ, s) is a vertex of K̂(M,a)k that maps to ϕ¯ under F̂ . It follows from the
definition of K̂(M)k that (ϕ¯, Ψ ◦ ρ, s) is automatically adjacent to (ϕ¯i,Ψi, si) for
i = 1, . . . ,m. This proves that F̂ has the link lifting property. This completes the
proof of the proposition.
82
8.4. Comparison with X•(M,a)k
We are now in a position to finally prove Theorem 8.0.3 by comparing
|X•(M,a)k| to |K̂(M,a)k|. We will need to construct an auxiliary semi-simplicial
space related to the simplicial complex K̂(M,a)k. Let M be a (4n + 1)-dimensional
manifold with non-empty boundary and let a : [0,∞)×R4n −→M be an embedding
as used in Definition 8.2.1. We define two semi simplicial spaces K̂•(M,a)k and
K̂ ′•(M,a)k.
Definition 8.4.1. The space of p-simplices K̂p(M,a)k is defined as follows:
i. The space of 0-simplices K̂0(M,a)k is defined to have the same underlying set
as the set of vertices of the simplicial complex K̂(M,a)k.
ii. The space of p-simplices K̂p(M,a)k ⊂ (K̂0(M,a)k)×(p+1) consists of the
ordered (p + 1)-tuples ((ϕ¯0,Ψ0, s0), · · · , (ϕ¯p,Ψp, sp)) such that the associated
unordered set
{(ϕ¯0,Ψ0, s0), · · · , (ϕ¯p,Ψp, sp)}
is a p-simplex in the simplicial complex K̂(M,a)k.
The spaces K̂p(M,a)k are topologized using the C
∞-topology on the spaces of
embeddings. The assignments [p] 7→ K̂p(M,a)k define a semi-simplicial space which
we denote by K̂•(M,a)k.
Finally, K̂ ′•(M,a)k ⊂ K̂•(M,a)k is defined to be the sub-semi-simplicial space
consisting of all (p + 1)-tuples ((ϕ¯0,Ψ0, s0), · · · , (ϕ¯p,Ψp, sp)) such that Ψi(W¯k) ∩
Ψj(W¯k) = ∅ whenever i 6= j.
It is easily verified that both K̂•(M,a)k and K̂ ′•(M,a)k are topological flag
complexes.
83
Proposition 8.4.1. Let k, n ≥ 2 be integers with k odd. Let M be a 2-connected
(4n + 1)-dimensional manifold and let g ≥ 0 be such that rk(M) ≥ g. Then the
geometric realization |K̂•(M,a)k| is 12(g − 4)-connected.
Proof. Let K̂•(M,a)δk denote the the discretization of K̂•(M,a)k as defined in
Definition 2.2.3. Consider the map
|K̂•(M,a)δk| −→ |K̂(M,a)k| (8.12)
induced by sending an ordered list ((ϕ¯0,Ψ0, s0), · · · , (ϕ¯p,Ψp, sp)) to its associated
underlying set. For any such set {(ϕ¯0,Ψ0, s0), · · · , (ϕ¯p,Ψp, sp)} which forms a
p-simplex in K̂(M,a)δk, there is only one possible ordering on it which yields an
element of K̂•(M,a)δk. Thus the map (8.12) is a homeomorphism. By Proposition
8.3.2, it follows that K̂•(M,a)δk (which is clearly a topological flag-complex) is
weakly Cohen-Macaulay of dimension 1
2
(g − 2), as defined in Definition 2.2.2. It
then follows from Theorem 2.2.1 that |K̂•(M,a)k| is 12(g − 4)-connected.
We now consider the inclusion map K̂ ′•(M,a)k −→ K̂•(M,a)k.
Proposition 8.4.2. For any (4n + 1)-dimensional manifold M with non-empty
boundary, the map |K̂ ′•(M,a)k| −→ |K̂•(M,a)k| induced by inclusion is a weak
homotopy equivalence.
Proof. For p ≥ 0, let
x 7→ ((ϕ¯x0 ,Ψx0 , sx0), · · · , (ϕ¯xp ,Ψxp , sxp)) for x ∈ Dj (8.13)
84
represent an element of the relative homotopy group
pij
(
K̂p(M,a)k, K̂
′
p(M,a)k
)
= 0. (8.14)
For each x, Yk(ϕ¯
x
i )
⋂
Yk(ϕ¯
x
j ) = ∅ whenever i 6= j. Using condition (c) in Definition
8.3.1, since Dj is compact we may choose a real number s ≥ max{sxi | i =
0, . . . , p, and x ∈ Dj}, such that for any x ∈ Dj,
Ψxi (W¯k, t) ∩Ψxj (W¯k, t) = ∅ whenever t ≥ s and i 6= j.
For each x ∈ Dj, t ∈ [0, 1], and i = 0, . . . , p, let sxi (t) denote the real number given
by the sum
(1− t) · sxi + t · s
and let Ψxi (t) denote the restriction of Ψ
x
i to W¯k × [sxi (t), ∞). The formula,
(x, t) 7→ ((ϕ¯x0 , Ψx0(t), sx0(t)), · · · , (ϕ¯xp , Ψxp(t), sxp(t))) for t ∈ [0, 1]
yields a homotopy from the map defined in (8.13) to a map which represents the
trivial element in the relative homotopy group (8.14). This implies that for all
p, j ≥ 0, the relative homotopy group (8.14) is trivial and thus the inclusion
K̂ ′p(M,a)k −→ K̂p(M,a)k is a weak homotopy equivalence for all p. It follows that
the induced map |K̂ ′•(M,a)k| −→ |K̂•(M,a)k| is a weak homotopy equivalence.
Finally, we consider the map
K̂ ′•(M,a)k −→ X•(M,a)k, (ϕ¯,Ψ, s) 7→ Ψs = Ψ|W¯k×{s}. (8.15)
85
The following proposition implies Theorem 8.0.3.
Proposition 8.4.3. Let n ≥ 2 and suppose that k > 2 is an odd integer. Then
for any (4n + 1)-dimensional manifold M with non-empty boundary, the degree
of connectivity of |X•(M,a)k| is bounded below by the degree of connectivity of
|K̂ ′•(M,a)k|.
Proof. To prove the proposition it will suffice to construct a section of the map
(8.15). The existence of such a section implies that the map on homotopy groups
induced by (8.15) is a surjection. The result then follows. Let x, y ∈ piτ2n(W¯k) be
two generators such that b(x, y) = 1
k
mod 1. By combining Corollary 5.4.2 and
Corollary 10.5.4, we may choose 〈k〉-embeddings ϕ0, ϕ1 : V 2n+1k −→M such that
[ϕ0β] = x, [ϕ
1
β] = y, and ϕ
0 t ϕ1 ∼= Ak.
We then may apply Construction 8.2.1 to obtain a vertex ϕ¯ = (ϕ,G, γ, t) ∈
K¯(W¯k, a)k. Now, the whole manifold W¯k is a regular neighborhood for Yk(ϕ¯). We
may choose a compression isotopy ρ : W¯k × [0,∞) −→ W¯k of W¯k to Yk(ϕ¯) as
in (8.11) and which satisfies the same conditions associated to the isotopy (8.11).
It follows that (ϕ¯, ρ, 0) is an element of K̂ ′0(W¯k, a)k. Using ϕ¯ and the compression
isotopy ρ, we then define a simplicial map
X•(M,a)k −→ K̂ ′•(M,a)k, Ψ 7→ (Ψ ◦ ϕ¯, Ψ ◦ ρ, 0), (8.16)
where Ψ ◦ ϕ¯ is the vertex in K¯(M,a)k given by the 4-tuple, ((Ψ ◦ ϕ0, Ψ ◦ ϕ1), Ψ ◦
G, Ψ ◦ γ, t). It follows that this map is a section of (8.15).
86
CHAPTER IX
HOMOLOGICAL STABILITY
With our main technical result Theorem 8.0.3 established, in this section we
show how Theorem 8.0.3 implies the main result of the paper which is Theorem
1.3.2.
9.1. A Model for BDiff∂(M)
Let M be a compact manifold of dimension m with non-empty boundary. We
now construct a concrete model for BDiff∂(M). Fix a collar embedding,
h : [0,∞)× ∂M −→M
with h−1(∂M) = {0}×∂M . Fix once and for all an embedding, θ : ∂M −→ R∞ and
let S denote the submanifold θ(∂M) ⊂ R∞.
Definition 9.1.1. We define M(M) to be the set of compact m-dimensional
submanifolds M ′ ⊂ [0,∞)× R∞ that satisfy:
i. M ′ ∩ ({0} × R∞) = S and M ′ contains [0, )× S for some  > 0.
ii. The boundary of M ′ is precisely {0} × S.
iii. M ′ is diffeomorphic to M relative to S.
Denote by E(M) the space of embeddings ψ : M → [0,∞) × R∞ for which there
exists  > 0 such that ψ ◦ h(t, x) = (t, θ(x)) for all (t, x) ∈ [0, ) × ∂M . The space
M(M) is topologized as a quotient of the space E(M) where two embeddings are
identified if they have the same image.
87
It follows from Definition 9.1.1 that M(M) is equal to the orbit space,
E(M)/Diff∂(M). By the main result of (4), the quotient map, E(M) −→
E(M)/Diff∂(M) = M(M) is a locally trivial fibre-bundle. This together with
the fact that E(M) is weakly contractible implies that there is a weak-homotopy
equivalence, M(M) ' BDiff∂(M).
Now suppose that m = 4n + 1 with n ≥ 2. Let k ≥ 2 be an integer. Recall
from Section I the manifold W˜k, given by forming the connected sum of [0, 1]× ∂M
with Wk. Choose a collared embedding α : W˜k −→ [0, 1] × R∞ such that for
(i, x) ∈ {0, 1} × ∂M ⊂ W˜k, the equation α(i, x) = (i, θ(x)) is satisfied. For any
submanifold M ′ ⊂ [0,∞) × R∞, denote by M ′ + e1 ⊂ [1,∞) × R∞ the submanifold
obtained by linearly translating M ′ over 1-unit in the first coordinate. Then for
M ′ ∈ M(M), the submanifold α(W˜k) ∪ (M ′ ∪ e1) ⊂ [0,∞) × R∞ is an element of
M(M ∪∂M W˜k). Thus, we have a continuous map,
sk :M(M) −→M(M ∪∂M W˜k); V 7→ α(W˜k) ∪ (V + e1). (9.1)
As in the introduction, we will refer to this map as the kth-stabilization map.
Remark 9.1.1. The construction of the stabilization map sk depends on the choice
of embedding α : W˜k → [0, 1] × R∞. However, any two such embeddings are
isotopic (the space of all such embeddings is weakly contractible). It follows that
the homotopy class of sk does not depend on any of the choices made. In this way,
the manifold W˜k determines a unique homotopy class of maps BDiff
∂(M) −→
BDiff∂(M ∪∂M W˜k) which is in the same homotopy class as the map (1.8) used
in the statement of Theorem 1.3.2.
88
9.2. A Semi-Simplicial Resolution
Let M be as in Section 9.1. For each positive integer K, we construct a semi-
simplicial space Z•(M)k, equipped with an augmentation k : Z•(M)k −→ M(M)
such that the induced map |Z•(M)k| −→ M(M) is highly connected. Such an
augmented semi-simplicial space is called a semi-simplicial resolution.
Let θ : ∂M ↪→ R∞ be the embedding used in the construction of M(M).
Pick once and for all a coordinate patch c0 : Rm−1 −→ S = θ(∂M). This choice
of coordinate patch induces for any M ′ ∈ M(M), a germ of an embedding [0, 1) ×
Rm−1 −→ M ′ as used in the construction of the semi-simplicial space K¯•(M ′)k from
Definition 4.1.1.
Definition 9.2.1. For each non-negative integer l, let Zl(M)k be the set of pairs
(M ′, φ¯) where M ′ ∈ M(M) and φ¯ ∈ Zl(M ′)k, where Xl(M ′)k is defined using the
embedding germ
[0, 1)× Rm−1 −→M ′
induced by the chosen coordinate patch c0 : Rm−1 −→ S. The space Zl(M)k
is topologized as the quotient, Zl(M)k = (E(M) × Xl(M)k)/Diff∂(M). The
assignments [l] 7→ Zl(M)k make Z•(M)k into a semi-simplicial space where the
face maps are induced by the face maps in X•(M)k.
The projection maps Zl(M)k −→ M(M) given by (V, φ¯) 7→ V yield an
augmentation map k : Zl(M)k −→ M(M). We denote by Z−1(M)k the space
M(M).
By construction, the projection maps Zl(M)k → M(M) are locally trivial
fibre-bundles with standard fibre given by Xl(M)k. From this we have:
89
Corollary 9.2.1. The map |k| : |Zl(M)k| −→ M(M) induced by the augmentation
is 1
2
(rk(M)− 2)-connected.
Proof. It follows from (31, Lemma 2.1) that there is a homotopy-fibre sequence
|Xl(M)k| → |Zl(M)k| → M(M). The result follows from the long-exact sequence on
homotopy groups.
9.3. Proof of the Main Theorem
We show how to use the semi-simplicial resolution k : Z•(M)k → M(M)
to complete the proof of Theorem 1.3.2. First, we fix some new notation which will
make the steps of the proof easier to state.
Let M be a compact (4n+ 1)-dimensional manifold with non-empty boundary.
Fix an odd integer k > 2 (the integer k will be same throughout the entire section).
For each g ∈ N we denote by Mg,k the manifold obtained by forming the connected-
sum of M with W#gk . Notice that ∂M = ∂Mg,k for all g ∈ N. We consider the
spaces M(Mg,k). For each g ∈ N, the stabilization map from (9.1) yields a map,
sk :M(Mg,k) −→M(Mg+1,k), M ′ 7→ W˜k ∪ (M ′ + e1).
Using the weak equivalence M(Mg,k) ' BDiff∂(Mg,k), Theorem 1.3.2 translates to
the following:
Theorem 9.3.1. The induced map (sk)∗ : Hl(M(Mg,k)) −→ Hl(M(Mg+1,k)) is an
isomorphism when l ≤ 1
2
(g − 3) and is an epimorphism when l ≤ 1
2
(g − 1).
We will need to consider the augmented semi-simplicial space Z•(Mg,k)k −→
M(Mg,k) that was constructed in the previous section.
90
Notational Convention 9.3.1. In order to prevent overcrowding in our notation,
thought the rest of this section we will drop k from the notation and denote
Mg := Mg,k and Z•(Mg) := Z•(Mg,k)k.
Since k is fixed throughout this section, this is not an issue.
Since rk(Mg) ≥ g for g ∈ N, it follows from Corollary 9.2.1 that the map
|k| : |Z•(Mg)| −→ Z−1(Mg) :=M(Mg).
is 1
2
(g − 2)-connected. With this established, the proof of Theorem 9.3.1 proceeds
in exactly the same way as in (9, Section 5). We provide an outline for how to
complete the proof and refer the reader to (9, Section 5) for details.
For what follows we fix g ∈ N. For each non-negative integer l ≤ g there is a
map
Fl :M(Mg−l−1) −→ Zl(Mg) (9.2)
which is defined in exactly the same way as the map from (9, Proposition 5.3).
From (9, Proposition 5.3, 5.4 and 5.5) we have the following.
Proposition 9.3.2. Let g ≥ 4.
i. The map Fl :M(Mg−l−1) −→ Zl(Mg) is a weak homotopy equivalence.
ii. The following diagram is commutative,
M(Mg−l−1)
Fl

sk //M(Mg−l)
Fl

Zl(Mg)
dl // Zl−1(Mg).
91
iii. The face maps di : Zl(Mg) −→ Zl−1(Mg) are weakly homotopic.
Remark 9.3.1. The proof of Proposition 9.3.2 proceeds in the same way as the
proofs of (9, Proposition 5.3, 5.4 and 5.5). The key ingredients of this proof are
Propositions 4.2.1 and 4.1.2.
We consider the spectral sequence E∗∗,∗ associated to the skeletal filtration of the
augmented semi-simplicial space
Z•(Mg) −→M(Mg).
This spectral sequence has the following properties:
– The E1-term given by E1p,q = Hq(Zp(Mg)) for l ≥ −1 and j ≥ 0.
– The differential is given by d1 =
∑
(−1)i(di)∗, where (di)∗ is the map on
homology induced by the ith face map in Z•(Mg).
– The group E∞p,q is a subquotient of the relative homology group
Hp+q+1(Z−1(Mg), |Z•(Mg)|).
Proposition 9.3.2 together with Corollary 9.2.1 imply the following:
(a) For g ≥ 4 + p, there are isomorphisms E1p,q ∼= Hq(M(Mg−p−1)).
(b) When p is even, the differential d1 : E1p,q −→ E1p−1,l is equal to (sk)∗ (the
map on homology induced by the k-th stabilization map) when p is even. It is
equal to zero when p is odd.
(c) The term E∞p,q is equal to 0 when p+ q ≤ 12(g − 4).
We now complete the proof of Theorem 9.3.1 by applying the spectral sequence
from (9, Proof of Theorem 1.2). We repeat the argument below for the convenience
of the reader.
92
Proof of Theorem 9.3.1. Let a denote the integer 1
2
(g − 4). We will use the above
spectral sequence to prove that Hq(Mg−1) −→ Hq(Mg) is an isomorphism for
q ≤ a, assuming that we know inductively that for j > 0, the stabilization maps
Hq(Mg−2j−1) −→ Hq(Mg−2j) are isomorphisms for q ≤ a− j.
This inductive assumption implies that the differential
d1 : E12j,q −→ E12j−1,q
is an isomorphism for 0 < j ≤ a − q. Hence, it follows that E2p,q = 0 whenever
0 < p ≤ 2(a− q). In particular the term E2p,q vanishes whenever
p ≥ 1, q ≤ a− 1, and p+ q ≤ a+ 1.
Thus for r ≥ 2 and q ≤ a, it follows that the differentials
dr : Err−1,q−r+1 −→ Er−1,q and dr : Err,q−r+1 −→ Er0,q
both vanish. It follows that for q ≤ a we have
E∞0,q = E
2
0,q = Ker(Hq(M(Mg−1))→ Hq(M(Mg)),
E∞−1,q = E
2
−1,q = Coker(Hq(M(Mg−1))→ Hq(M(Mg)).
Since the group E∞p,q vanishes for p+ q ≤ a we see that the stabilization map
Hq(M(Mg−1)) −→ Hq(M(Mg))
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has vanishing kernel and cockerel for q ≤ a, establishing the induction step. The
statement of the theorem is vacuous for g = 1 and g = 2, which establishes the
base case of the induction. A similar argument proves that the stabilization map
Hq(M(Mg−1)) −→ Hq(M(Mg)) is an epimorphism when q = a + 1. This completes
the proof of the theorem.
94
CHAPTER X
EMBEDDINGS AND DISJUNCTION FOR Z/K-MANIFOLDS
We now develop a technique for modifying the intersections of embedded 〈k〉-
manifolds that will allow us to prove Theorem 7.4.1 stated in Section VII. Recall
from Section 7.4 the definition of diffeotopy.
Definition 10.0.1. Let M be a manifold. We will call a smooth, one parameter
family of diffeomorphisms Ψt : M −→ M with t ∈ [0, 1] and Ψ0 = IdM a diffeotopy.
For a subspace N ⊂ M , we say that Ψt is a diffeotopy relative N , and we write
Ψt : M −→M rel N , if in addition, Ψt|N = IdN for all t ∈ [0, 1].
10.1. A Z/k Version of the Whitney Trick
We now discus a certain version of the Whitney trick for 〈k〉-manifolds.
Let M be an oriented manifold of dimension m, let X be an oriented manifold of
dimension r, and let P be an oriented 〈k〉-manifold of dimension p. Suppose that:
– both P and X are path-connected,
– m ≥ 6,
– p+ r = m,
– p, r ≥ 2.
Let
ϕ : (X, ∂X) −→ (M,∂M) and f : (P, ∂0P ) −→ (M,∂M) (10.1)
95
be a smooth embedding and a 〈k〉-embedding respectively, such that
ϕ(∂X) ∩ f(∂0P ) = ∅.
We will need to consider the invariant Λ0k(f, ϕ;M) defined in Section VII. Using the
standard identification
ΩSO0 (pt.)〈k〉 = Z/k,
the element Λ0k(f, ϕ;M) is equal to the modulo k reduction of the oriented,
algebraic intersection number associated to the intersection of f(Int(P )) and ϕ(X).
The following theorem is a version of the classical Whitney trick for 〈k〉-manifolds.
Theorem 10.1.1. Let f and ϕ be exactly as in (10.1) above. Using the
identification ΩSO0 (pt.)〈k〉 = Z/k, suppose that
Λ0k(f, ϕ;M) = j mod k.
Then there exists a diffeotopy Ψt : M −→M rel ∂M such that,
Ψ1(ϕ(X)) ∩ f(Int(P )) ∼= +〈j〉.
To prove the above theorem we will need to use the next lemma.
Lemma 10.1.2. Let P be a 〈k〉-manifold of dimension p ≥ 2, let M be a smooth
manifold of dimension m ≥ 6, and let f : (P, ∂0P ) −→ (M,∂M) be a 〈k〉-
embedding. Let r denote the integer m − p. Given any any positive integer n, there
exists an embedding
g : Sr −→ Int(M \ fβP (βP ))
96
that satisfies the following:
i. g(Sr) ∩ f(Int(P )) ∼= ±〈n · k〉,
ii. the composition Sr
g−→ Int(M \ fβP (βP )) ↪→ Int(M) extends to an embedding
Dr+1 ↪→ Int(M).
Proof. We first prove this explicitly for the case that n = 1. So, suppose that n is
equal to 1. We construct the embedding g : Sr −→ Int(M) in stages as follows.
Construction 10.1.1. Let Φ¯ : ∂1P −→ βP be the map given by the composition
∂1P
Φ
∼=
// βP × 〈k〉 proj. // βP.
For i = 1, . . . , k, let ∂i1P denote the submanifold given by Φ
−1(∂1P × {i}).
i. Choose a collar embedding h : ∂1P × [0,∞) −→ P such that h−1(∂1P ) =
∂1P × {0}.
ii. Choose a point y ∈ βP . For i = 1, . . . , k, let yi ∈ ∂i1P be the point such that
Φ¯(yi) = y. Then define embeddings
γi : [0, 1] −→ f(P ), γi(t) = f(h(yi, t)).
It is clear that γi(0) = f(y) for all i. We then denote
xi := γi(1) for i = 1, . . . , k.
iii. Choose an embedding α : D2 −→M that satisfies the following conditions:
97
(a) α(D2) ∩ f(P ) = unionsqki=1γi([0, 1]),
(b) α(∂D2) ∩ f(P ) = {x1, . . . , xk},
(c) α(D2) intersects f(P ) orthogonally (with resect to some metric on M),
(d) f(βP ) ∩ α(D2) ⊂ α(Int(D2)).
Since 2 < m/2, there is no obstruction to choosing such an embedding.
iv. Let r denote the integer m − p. Choose a (r − 1)-frame of orthogonal vector
fields (v1, . . . , vr−1) over the embedded disk α(D2) ⊂ M with the property
that vi is orthogonal to α(D
2) and orthogonal to f(P ) over the intersection
α(D2) ∩ f(P ), for i = 1, . . . , r − 1. Since the disk is contractable, there is no
obstruction to the existence of such a frame.
The orthogonal (r − 1)-frame chosen in step iv. induces an embedding
g¯ : Dr+1 −→M.
The orthogonality condition (condition (c)) in Step iii. of the above construction,
together with the orthogonality condition on the frame chosen in step iv., implies
that g¯(Dr+1) is transverse to f(P ). Furthermore, condition (b) from step iii. of the
above construction implies that
g(∂Dr+1) ∩ f(Int(P )) = {x1, . . . , xk},
and all points on the right-hand side of the equality have the same orientation. We
then set the map g : Sr −→ M equal to the embedding obtained by restricting g¯
to the boundary of Dr+1. This proves the lemma in the case that n = 1. To prove
98
the lemma for general n one simply iterates n-times the exact construction given
above.
Proof of Theorem 10.1.1. It will suffice to prove the following: suppose that
f(Int(P )) ∩ X consists of exactly k points, all of which are positively oriented.
Then there exists a diffeotopy Ψt : M −→ M rel ∂M such that Ψ0 = IdM and
Ψ1(X)∩f(P ) = ∅. So, suppose that f(Int(P ))∩X consists of exactly k points, all of
which are positively oriented. By the previous lemma, there exists and embedding
g : Sr −→ Int(M \X ∪ fβ(βP ))
such that g(Sr) ∩ f(Int(P )) consists of exactly k points, all of which are negatively
oriented. Furthermore, the embedding g can be chosen so that it admits an
extension to an embedding
g¯ : Dr+1 −→ Int(M \X).
Let X˜ ⊂ M be the submanifold obtained by forming the connected sum of g(Sr) ⊂
M with X along some embedded arc in M that is disjoint from f(P ). It follows
easily from the fact that g extends to an embedding of a disk, that X˜ is ambient
isotopic to X. By construction, it follows that we have
f(Int(P )) ∩ X˜ ∼= +〈k〉 unionsq −〈k〉.
99
Since both f(Int(P )) and X˜ are path connected and M is simply connected by
assumption, we may then apply the Whitney trick to obtain a diffeotopy
Ψt : M −→M rel ∂M
with Ψ1(X) ∩ f(P ) = ∅. This concludes the proof of the theorem.
10.2. A Higher Dimensional Intersection Invariant
We recall now a certain construction developed by Hatcher and Quinn in (16).
Let M , X, and Y be smooth manifolds of dimension m, r, and s respectively. Let
t = r + s−m. Let
ϕ : (X, ∂X) −→ (M,∂M) and ψ : (Y, ∂Y ) −→ (M,∂M)
be smooth maps. Let E(ϕ, ψ) denote the homotopy pull-back of ϕ and ψ.
Specifically, this is given by
E(ϕ, ψ) = {(x, y, γ) ∈ X × Y × Path(M) | ϕ(x) = γ(0), ψ(y) = γ(1) }.
Consider the diagram,
E(ϕ, ψ)
piX //
piM
((
piY

X
ϕ

Y
ψ //M
(10.2)
where piX and piY are projection maps and piM is given by the formula (x, y, γ) 7→
γ(1/2). It is easily verified that this diagram commutes up to homotopy. Let νX
and νY denote the stable normal bundles associated to X and Y . We will need to
100
consider the stable vector bundle over E(ϕ, ψ) given by the Whitney sum
pi∗X(νX)⊕ pi∗Y (νY )⊕ pi∗M(TM).
We will denote this stable bundle by ν̂(ϕ, ψ). We will need to consider the normal
bordism group
Ωfr.t (E(ϕ, ψ), ν̂(ϕ, ψ)).
Elements of this bordism group are represented by triples (N, f, F ), where N is a
t-dimensional closed manifold, f : N −→ E(ϕ, ψ) is a map, and F : νN −→ ν̂(ϕ, ψ)
is an isomorphism of stable vector bundles covering the map f .
Now, suppose that the maps ϕ and ψ are transversal and that
ϕ(∂X) ∩ ψ(∂Y ) = ∅.
It follows that the pullback ϕ t ψ ⊂ X × Y is a closed submanifold of dimension t.
There is a natural map
ιϕ,ψ : ϕ t ψ −→ E(ϕ, ψ), (x, y) 7→ (x, y, cϕ(x)),
where cϕ(x) is the constant path at point ϕ(x). Let νϕtψ denote the stable normal
bundle associated to the pull-back ϕ t ψ. The following is given in (16, Proposition
2.1) (see also the discussion on Pages 331-332).
Proposition 10.2.1. There is a natural bundle isomorphism ιˆϕ,ψ : νϕtψ
∼=−→
ν(ϕ, ψ), determined uniquely by the homotopy classes of ϕ and ψ, that covers the
map ιϕ,ψ. In this way, the triple (ϕ t ψ, ιϕ,ψ, ιˆϕ,ψ) determines a bordism class in
Ωfr.t (E(ϕ, ψ), ν̂(ϕ, ψ)).
101
The bordism group Ωfr.t (E(ϕ, ψ), ν̂(ϕ, ψ)) can be quite difficult to compute
in general. However, in the case that the manifolds X, Y , and M are highly
connected, the group Ωfr.t (E(ϕ, ψ), ν̂(ϕ, ψ)) reduces to something much more
simple. The following proposition is proven in (16, Section 3).
Proposition 10.2.2. Suppose that X, Y , and M are (t + 1)-connected (recall that
t = dim(X) + dim(Y )− dim(M) = r + s−m). Then the homomorphism
Ωfr.t (pt.)→ Ωfr.t (E(ϕ, ψ), ν̂(ϕ, ψ))
induced by the inclusion of any point into E(ϕ, ψ), is an isomorphism.
Definition 10.2.1. In the case that X, Y , and M are (t + 1)-connected, we will
denote by
αt(ϕ, ψ;M) ∈ Ωfr.t (pt.) (10.3)
the image of the bordism class in Ωfr.t (E(ϕ, ψ), ν̂(ϕ, ψ)) associated to ϕ t ψ under
the isomorphism of the previous proposition.
Remark 10.2.1. We emphasize that it is not necessary for both ϕ and ψ to be
embeddings in order for the class αt(ϕ, ψ;M) to be defined. It is only necessary
that ϕ and ψ be transversal as smooth maps. Furthermore its is easy to see that
αt(ϕ, ψ,M) is an invariant of the homotopy class of ϕ and ψ. However, the next
theorem (Theorem 10.2.3) does require that ϕ and ψ be embeddings.
The following is proven in (16, Theorem 2.2) (and in (36)).
Theorem 10.2.3. Let
ϕ : (X, ∂X) −→ (M,∂M) and ψ : (Y, ∂Y ) −→ (M,∂M)
102
be embeddings such that ϕ(∂X) ∩ ψ(∂Y ) = ∅. Suppose that m > r + s
2
+ 1 and
m > s+ r
2
+1, and that X, Y , and M are (t+1)-connected. Then if αt(ϕ, ψ;M) = 0,
there exists a diffeotopy
Ψt : M −→M rel ∂M
such that Ψ1(ϕ(X)) ∩ ψ(Y ) = ∅.
Remark 10.2.2. In (16) the above theorem is only explicitly proven in the case
when X and Y are closed manifolds, though their proof can easily by modified to
yield the version stated above. In (41), a proof of the relative version stated exactly
as above is given.
The next lemma, which we will use latter, is a restatement of (16, Theorem
1.1).
Lemma 10.2.4. Let
ϕ : (X, ∂X) −→ (M,∂M) and ψ : (Y, ∂Y ) −→ (M,∂M)
be embeddings with ϕ(∂X) ∩ ψ(∂Y ) = ∅. Suppose that ϕ is homotopic relative ∂X,
to a map ϕ′ such that ϕ′(X) ∩ ψ(Y ) = ∅. If m > r + s/2 + 1, then there exists a
diffeotopy
Ψt : M −→M rel ∂M
such that (Ψ1 ◦ ϕ(X)) ∩ ψ(Y ) = ∅.
Remark 10.2.3. The main dimensional case when will use Theorem 10.2.3 and
Lemma 10.2.4 is when dim(M) = 2n+ 1, dim(X) = dim(Y ) = n+ 1, and n ≥ 4.
103
10.3. Creating Intersections
There is a particular application of the above theorem that we will need
to use. Let M and Y be oriented, connected manifolds of dimension m and s
respectively and let
ψ : (Y, ∂Y ) −→ (M,∂M)
be an embedding. Let r = m − s and let ϕ : Sr −→ Int(M) be a smooth map
transverse to ψ(Y ) ⊂ M . Let j ≥ 0 be an integer strictly less than r and let
γ : Sr+j −→ Sr be a smooth map. Denote by
Pj : pir+j(Sr)
∼=−→ Ωfr.j (pt.) (10.4)
the Pontryagin-Thom isomorphism (see (23)). The following lemma shows how to
compute
αj(ϕ ◦ γ, ψ; M)
in terms of α0(ϕ, ψ,M) and the element Pj([γ]) ∈ Ωfr.j (pt.).
Lemma 10.3.1. Let ψ, ϕ and γ : Sr+j → Sr be exactly as above. Then
αj(ϕ ◦ γ, ψ; M) = α0(ϕ, ψ;M) · Pj([γ]),
where the product on the right-hand side is the product in the graded bordism ring
Ωfr.∗ (pt.).
Proof. Let s ∈ Z denote the oriented, algebraic intersection number associated to
the intersection of ϕ(Sr) and ψ(Y ). By application of the Whitney trick, we may
104
deform ϕ so that
ϕ(Sr) ∩ ψ(Y ) = {x1, . . . , x`}, (10.5)
where the points xi for i = 1, . . . , ` all have the same sign. It follows that
(ϕ ◦ γ)−1(ψ(Y )) =
⊔`
i=1
γ−1(xi).
For each i ∈ {1, . . . , `}, the framing at xi (induced by the orientations of γ(Sr),
ψ(Y ) and M) induces a framing on γ−1(xi). We denote the element of Ωfr.1 (pt.)
given by γ−1(xi) with this induced framing by [γ−1(xi)]. By definition of the
Pontryagon-Thom map Pj (see (23, Section 7)), the element [γ−1(xi)] is equal to
Pj([γ]) for i = 1, . . . , `. Using the equality (10.5), it follows that
αj(ϕ ◦ γ, ψ; M) = ` · Pj([γ]).
The proof then follows from the fact that α0(ϕ, ψ,M) is identified with the
algebraic intersection number associated to ϕ(Sr) and ψ(Y ).
10.4. A Technical Lemma
Before we proceed further, we develop a technical result that will play an
important role in the proof of Theorem 10.5.1. For n ≥ 4, let M be a 2-connected,
oriented (2n + 1)-dimensional manifold and let P be a 2-connected, oriented, 〈k〉-
manifold of dimension n+1. Let f : (P, ∂0P ) −→ (M,∂M) be a 〈k〉-embedding. Let
U be a tubular neighborhood of fβ(βP ) ⊂ M whose boundary intersects f(Int(P ))
transversally. Denote,
Z := M \ Int(U), P ′ := f−1(Z), f ′ := f |P ′ . (10.6)
105
It follows from the fact that ∂U intersects Int(f(P )) transversally that P ′ is a
smooth manifold with boundary (after smoothing corners) and that f ′ maps ∂P ′
into ∂M . Let ξ denote the generator of the framed bordism group Ωfr.1 (pt.), which
is isomorphic to Z/2.
Lemma 10.4.1. Let f : (P, ∂0P ) −→ (M,∂M) be as above and let iZ : Z ↪→
M denote the inclusion map. There exists an embedding ϕ : Sn+1 −→ Z which
satisfies:
i. α1(f
′, ϕ;Z) = k · ξ ∈ Ωfr.1 (pt.),
ii. the composition iZ ◦ ϕ : Sn+1 −→M is null-homotopic.
Proof. By Lemma 10.1.2, we may choose an embedding φ : Sn −→ M \ fβ(βP ) an
embedding that satisfies:
– φ(Sn) intersects f(Int(P )) transversally,
– φ(Sn) ∩ f(Int(P )) ∼= +〈k〉,
– iZ ◦ φ : Sn →M extends to an embedding Dn+1 ↪→M .
By shrinking the tubular neighborhood U of fβ(βP ) if necessary, we may assume
that
φ(Sn) ⊂ Z = M \ Int(U).
Denote by φ̂ : Sn −→ Z the map obtained by restricting the codomain of φ. Let
γ : Sn+1 −→ Sn
106
represent the generator of pin+1(S
n) ∼= Z/2. By Lemma 10.3.1 it follows that,
α1(φ̂ ◦ γ, f ′;Z) = α0(φ̂, f ′;Z) · P1([γ]) = k · P1([γ]) = k · ξ,
where P1 : pin+1(Sn) −→ Ωfr.1 (pt.) is the Pontryagin-Thom map for framed bordism.
Since Z is 2-connected and n ≥ 4, we may apply (37, Proposition 1) (or the
main theorem of (12)), and find a homotopy of the map φ̂ ◦ γ, to an embedding
ϕ : Sn+1 −→ Z. Since the map iZ ◦ φ : Sn −→ M is null-homotopic, it follows
that iZ ◦ ϕ : Sn+1 → M is null-homotopic as well. This completes the proof of the
lemma.
10.5. Modifying Intersections
We now state the main result of this section (which is a restatement of
Theorem 7.4.1 from Section 7.4). Fix an integer n ≥ 4, let M be an oriented, 2-
connected manifold of dimension 2n + 1. Let P and Q be oriented, 2-connected,
〈k〉-manifolds of dimension n + 1. We will use the same M , P , and Q throughout
this section.
Theorem 10.5.1. With M , P , and Q as above and let
f : (P, ∂0P ) −→ (M,∂M) and g : (Q, ∂0Q) −→ (M,∂M)
be transversal 〈k〉-embeddings such that f(∂0P ) ∩ g(∂0Q) = ∅. Suppose that
Λ1k,k(f, g;M) = 0. If the integer k is odd, then there exists a diffeotopy Ψt : M −→
M rel ∂M such that Ψ1(f(P )) ∩ g(Q) = ∅.
107
The proof of the above theorem is proven in stages via several intermediate
propositions.
Proposition 10.5.2. Let
f : (P, ∂0P ) −→ (M,∂M) and g : (Q, ∂0Q) −→ (M,∂M)
be 〈k〉-embeddings as above and suppose that
β1(Λ
1
k,k(f, g;M)) = Λ
0
k(fβ, g;M) = 0.
Then there exists a diffeotopy Ψt : M −→ M rel ∂M such that Ψ1(fβ(βP )) ∩ g(Q) =
∅.
Proof. Since 0 = β1(Λ
1
k,k(f, g;M)) = Λ
0
k(g, fβ;M), it follows that the algebraic
intersection number associated to fβ(βP ) and g(IntQ) is a multiple of k. The
desired diffeotopy exists by Theorem 10.1.1.
Proposition 10.5.3. Let g : (Q, ∂0Q) −→ (M,∂M) be a 〈k〉-embedding as above.
Let X be a smooth manifold of dimension n + 1 and let ϕ : (X, ∂X) −→ (M,∂M)
be a smooth embedding such that
ϕ(∂X) ∩ g(∂0Q) = ∅.
If the integer k is odd, then there exists a diffeotopy, Ψt : M → M rel ∂M such
that,
Ψ1(ϕ(X)) ∩ g(Q) = ∅.
108
Proof. By Proposition 7.2.1, we have
β(Λ1k(g, ϕ;M)) = Λ
0(gβ, ϕ;M) ∈ ΩSO0 (pt.)
where
β : ΩSO1 (pt.)〈k〉 −→ ΩSO0 (pt.), [V ] 7→ [βV ]
is the Bockstein homomorphism. By (5.2), this Bockstein homomorphism is
the zero map for all k (the group ΩSO1 (pt.)〈k〉 is equal to zero). It follows that
Λ0(gβ, ϕ;M) ∈ ΩSO0 (pt.) is the zero element and thus the oriented, algebraic
intersection number associated to gβ(βQ) ∩ X is equal to zero. By application of
the Whitney trick (24, Theorem 6.6), we may find a diffeotopy of M , relative ∂M ,
which pushes X off of the submanifold gβ(βQ) ⊂ M . Using this, we may now
assume that ϕ(X) ∩ g(∂1Q) = ∅.
Let U ⊂ M be a closed tubular neighborhood of fβ(βP ), disjoint from X,
such that the boundary of U intersects f(P ) transversely. As in (10.6), we denote
Z := M \ IntU, P ′ := f−1(Z), f ′ := f |P ′ .
Notice that P ′ is a manifold with boundary and that f ′ is an embedding which
maps (P ′, ∂P ′) into (Z, ∂Z). Furthermore, ϕ maps (X, ∂X) into (Z, ∂Z). To prove
the corollary it will suffice to construct a diffeotopy Ψ′t : Z −→ Z rel ∂Z such
that Ψ′1(X) ∩ P ′ = ∅. By Theorem 10.2.3, the obstruction to the existence of such
a diffeotopy is the class α1(f
′, ϕ;Z) ∈ Ωfr1 (pt.). If α1(f ′, ϕ;Z) is equal to zero, we
are done. So suppose that α1(f
′, ϕ;Z) = ξ where ξ is the non-trivial element in
109
Ωfr1 (pt.)
∼= Z/2. Denote by iZ : Z ↪→ M the inclusion map. By Lemma 10.4.1 there
exits an embedding φ : Sn+1 −→ Z such that:
– α1(f
′, φ;Z) = k · ξ where ξ ∈ Ωfr1 (pt.) ∼= Z/2 is the standard generator,
– the embedding iZ ◦ φ : Sn+1 −→M is null-homotopic.
Since k is odd, we have α1(f
′, φ;Z) = ξ. We denote by ϕ̂ : X −→ M the
embedding obtained by forming the connected sum of ϕ(X) with iZ ◦ ϕ(Sn+1) along
the thickening of an embedded arc that is disjoint from f(P ), U , and X. Since
iZ ◦ ϕ : Sn+1 −→ M is null-homotopic, it follows that ϕ̂ is homotopic, relative to
∂X, to the original embedding ϕ. We have
α1(f
′, ϕ̂;Z) = α1(f ′, ϕ;Z) + α1(f ′, φ;Z) = ξ + ξ = 0,
and so there exists a diffeotopy Ψ′t : Z → Z rel ∂Z such that Ψ′1(ϕ̂(X))∩ f ′(P ′) = ∅.
We then extend Ψ′t identically over M \ Z to obtain a diffeotopy
Ψ̂t : M −→M rel ∂M
such that Ψ̂1(ϕ̂(X)) ∩ f(P ) = ∅. Now, since ϕ is homotopic relative ∂X to the
embedding Ψ̂1 ◦ ϕ̂ and Ψ̂1(ϕ̂(X))∩ f(P ) = ∅, we may apply Lemma 10.2.4 to obtain
a diffeotopy
Ψt : M −→M rel ∂M
such that (Ψ1 ◦ ϕ(X)) ∩ f(P ) = ∅. This concludes the proof of the proposition.
We can now complete the proof of Theorem 10.5.1.
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Proof of Theorem 10.5.1. By hypothesis we have Λ1k,k(f, g;M) = 0, and thus
Λ0k(fβ, g;M) = 0, and so by Proposition 10.5.2 we may assume that fβ(βP ) ∩
g(Q) = ∅. Choose a closed tubular neighborhood U ⊂ M of fβ(βP ), disjoint from
g(Q), with boundary transverse to f(P ). As in (10.6) we denote,
Z := M \ IntU, P ′ := f−1(Z), and f ′ := f |P ′ . (10.7)
With these definitions, P ′ is an oriented manifold with boundary and
f ′ : (P ′, ∂P ′) −→ (Z, ∂Z)
is an embedding. Furthermore, since U was chosen to be disjoint from g(Q), we
have g(Q) ⊂ Z. Let g′ : (Q, ∂0) −→ (Z, ∂Z) denote the 〈k〉-embedding obtained by
restricting the codomain of g. To finish the proof, we then apply Proposition 10.5.3
to the embedding f ′ : (P ′, ∂P ′) −→ (Z, ∂Z) and 〈k〉-embedding g′ : (Q, ∂0Q) −→
(Z, ∂Z), to obtain a diffeotopy of Z (relative ∂Z) that pushes f ′(P ′) off of g′(Q).
This completes the proof of the theorem.
We now come to an important corollary. Recall from Section 6.3 the 〈k, k〉-
manifold Ak.
Corollary 10.5.4. Let f and g be exactly as in the statement of Theorem 10.5.1.
Suppose that the class Λ1k,k(f, g;M) is equal to the class represented by the closed
1-dimensional 〈k, k〉-manifold +Ak. If k is odd then there exists a diffeotopy Ψt :
M −→ M rel ∂M such that the transverse pull-back (Ψ1 ◦ f) t g is diffeomorphic to
Ak.
111
Proof. Since Λ1k,k(f, g;M) is equal to the class represented by +Ak in Ω
SO
1 (pt.)〈k,k〉,
it follows that f t g is diffeomorphic (as an oriented 〈k, k〉-manifold) to the
disjoint union of precisely one copy of +Ak together with some other oriented
〈k, k〉-manifold, that represents the zero element in ΩSO1 (pt.)〈k,k〉. We may write
f(P ) ∩ g(Q) = A unionsq Y, (10.8)
where
A ∼= +Ak and [Y ] = 0 in ΩSO1 (pt.)〈k,k〉.
Let U ⊂M be a closed neighborhood of fβ(βP )∪A, disjoint from Y , with boundary
transverse to both f(P ) and g(Q). We then denote
Z := M \ Int(U), P ′ := f−1(Z), Q′ := g−1(Z). (10.9)
Notice that both P ′ and Q′ are 〈k〉-manifolds with
∂0P
′ = f−1(∂Z), ∂1P ′ = (f |∂1P )−1(Z), βP ′ = f−1β (Z),
∂0Q
′ = g−1(∂Z), ∂1Q′ = (g|∂1Q)−1(Z), βQ′ = g−1β (Z).
We denote by
f ′ : (P ′, ∂0P ′) −→ (Z, ∂Z) and g′ : (Q′, ∂0Q′) −→ (Z, ∂Z)
the 〈k〉-embeddings given by restricting f and g. By construction, the pull-back
f ′ t g′ is diffeomorphic as an oriented 〈k, k〉-manifold to Y , which represents the
zero element in ΩSO1 (pt.)〈k,k〉. It follows that Λ
1
k,k(f
′, g′;Z) = 0. By Theorem 10.5.1
112
we obtain a diffeotopy Ψt : Z −→ Z rel ∂Z, such that Ψ1(f ′(P ′)) ∩ g′(Q′) = ∅. This
concludes the proof.
113
CHAPTER XI
IMMERSIONS AND EMBEDDINGS OF Z/K-MANIFOLDS
In this section we determine the conditions for when a 〈k〉-map can be
deformed to a 〈k〉-immersion or a 〈k〉-embedding. The techniques of this section
enable us to prove Theorem 5.4.1 (which is restated again in this section as
Theorem 11.6.1).
11.1. A Recollection of Smale-Hirsch Theory
Let N and M be smooth manifolds of dimensions n and m respectively.
Denote by Imm(N,M) the space of immersions N → M , topologized in the
C∞-topology. Let Immf (N,M) denote the space of bundle maps TN −→ TM
which are fibre-wise injective. Elements of the space Immf (N,M) are called formal
immersions. There is a map D : Imm(N,M) −→ Immf (N,M) defined by sending
an immersion φ : N −→ M to the bundle injection given by its differential
Dφ : TN −→ TM . The following theorem is proven in (1, Chapter III, Section
9) and is originally due to Hirsch and Smale.
Theorem 11.1.1. The if dim(N) < dim(M), then the map D : Imm(N,M) −→
Immf (N,M) is a weak homotopy equivalence. In the case that dim(N) = dim(M),
then D is a weak homotopy equivalence if N is an open manifold.
Let Îmm(N,M) denote the space of pairs (φ,v) ∈ Imm(N,M)×Maps(N, TM) that
satisfy:
i. pi(v(x)) = φ(x) for all x ∈ N , where pi : TM →M is the bundle projection,
114
ii. for each x ∈ N , the vector v(x) is transverse to the vector subspace
Dφ(TxN) ⊂ Tφ(x)M,
where Dφ is the differential of φ.
Similarly, we define Îmm
f
(N,M) to be the space of pairs (ψ,v) ∈ Immf (N,M) ×
Maps(N, TM) which satisfy:
i. pi(v(x)) = pi(ψ(x)) for all x ∈ N , where pi : TM → M is the bundle
projection,
ii. for all x ∈ N , the vector v(x) is transverse to the vector subspace
ψ(TxN) ⊂ Tpi(ψ(x))M.
There is a map
D̂ : Îmm(N,M) −→ Îmmf (N,M), (φ,v) 7→ (Dφ,v). (11.1)
The following is an easy corollary of Theorem 11.1.1.
Corollary 11.1.2. Suppose that dim(N) < dim(M). Then the map D̂ from (11.1)
is a weak homotopy equivalence.
11.2. The Space of 〈k〉-Immersions
We now proceed to prove a version of Theorem 11.1.2 for immersions of 〈k〉-
manifolds. For what follows, let M be a manifold of dimension m and let P be
115
a 〈k〉-manifold of dimension p. We will need to construct a suitable space of 〈k〉-
immersions and formal 〈k〉-immersions.
Choose a collar embedding h : ∂1P×[0,∞) −→ P , with h−1(∂1P ) = ∂1P×{0}.
Denote by vh ∈ Γ∂1P (TP ) the inward pointing vector field along ∂1P determined by
the differential of the collar embedding h. Using vh we have maps,
R : Imm(P,M) −→ Îmm(∂1P,M), φ 7→ (φ|∂P , Dφ ◦ vh),
Rf : Immf (P,M) −→ Îmmf (∂1P,M), ψ 7→ (ψ|∂P , ψ ◦ vh).
(11.2)
The next lemma follows from the basic results of (1, Chapter III: Section 9).
Lemma 11.2.1. The map Rf is a Serre-fibration in the case that dim(P ) ≤
dim(M). The map R is a Serre-fibration in the case that dim(P ) < dim(M).
Let Φ¯ : ∂1P −→ βP be the map given by the composition
∂1P
Φ
∼=
// βP × 〈k〉projβP // βP. Using Φ¯ we have a map
Tk : Îmm(βP,M) −→ Îmm(∂1P,M), (φ,v) 7→ (φ ◦ Φ¯, v ◦ Φ¯). (11.3)
Similarly, by using the differential DΦ¯ of Φ¯, we define a map
T fk : Îmm
f
(βP,M) −→ Îmmf (∂1P,M), (ψ,v) 7→ (ψ ◦DΦ¯, v ◦ Φ¯). (11.4)
Definition 11.2.1. We define Imm〈k〉(P,M) to be the space of pairs
(φ, (φ′,v)) ∈ Imm(P,M)× Îmm(βP,M)
116
such that Tk(φ
′,v) = R(φ). Similarly we define Immf〈k〉(P,M) to be the space of
pairs
(ψ, (ψ′,v)) ∈ Immf (P,M)× Îmmf (βP,M)
such that T fk (ψ
′,v) = Rf (ψ).
Remark 11.2.1. Let (φ, (φ′,v)) ∈ Imm〈k〉(P,M). By construction, the
immersion φ : P −→ M is a 〈k〉-immersion and φ′ = φβ. The pair (φ′,v) is
completely determined by the 〈k〉-immersion φ and so, the space Imm〈k〉(P,M) is
homeomorphic to the subspace of Maps〈k〉(P,M) consisting of all 〈k〉-immersions
P →M .
Lemma 11.2.2. The following two commutative diagrams
Imm〈k〉(P,M) //

Imm(P,M)
R

Immf〈k〉(P,M)

// Immf (P,M)
Rf

Îmm(βP,M)
Tk // Îmm(∂1P,M), Îmm
f
(βP,M)
T fk // Îmm
f
(∂1P,M),
are homotopy cartesian.
Proof. This follows immediately from Lemma 11.2.1 and the fact that both of the
diagrams are pull-backs.
Finally we may consider the map
Dk : Îmm〈k〉(P,M) −→ Îmm
f
〈k〉(P,M), (φ, (φ
′,v)) 7→ (Dφ, (Dφ′,v)). (11.5)
We have the following theorem.
Theorem 11.2.3. Suppose that dim(P ) < dim(M). Then the map Dk of (11.5) is
a weak homotopy equivalence.
117
Proof. The map from (11.5) induces a map between the two commutative squares
in Lemma 11.2.2. The maps between the entries on the bottom row and the
entries on the upper-right are weak homotopy equivalences by Theorem 11.1.1 and
Corollary 11.1.2. It then follows from Lemma 11.2.2 that the upper-left map (which
is (11.5)) is a weak homotopy equivalence.
11.3. Representing Classes of 〈k〉-Maps by 〈k〉-Immersions
Let P be a 〈k〉-manifold of dimension p and let h : ∂1 × [0,∞) −→ P be a
collar embedding with h−1(∂1P ) = ∂1P × {0}. We have a bundle map
Φ∗ : TP |∂1P −→ T (βP )⊕ 1 (11.6)
given by the composition, TP |∂1P
∼= // T (∂1P )⊕ 1
DΦ¯⊕Id1 // T (βP )⊕ 1, where
the first map is the bundle isomorphism induced by the collar embedding h. Using
this bundle isomorphism Φ∗, we define a new space T P̂ as a quotient of TP by
identifying two points v, v′ ∈ TP |∂1P ⊂ TP if and only if Φ∗v = Φ∗v′. With this
definition, there is a natural projection ̂ : T P̂ −→ P̂ which makes the diagram
TP
pi

// T P̂
̂
P // P̂
(11.7)
commute. It is easy to verify that the projection map ̂ : T P̂ −→ P̂ is a vector
bundle and that the upper-horizontal map in the above diagram is a bundle map
that is an isomorphism on each fibre.
Definition 11.3.1. The 〈k〉-manifold P is said to be parallelizable if the induced
vector bundle ̂ : T P̂ → P̂ is trivial.
118
Corollary 11.3.1. Let P be a parallelizable 〈k〉-manifold and let M be a manifold
of dimension greater than dim(P ). Let f : P −→ M be a 〈k〉-map and consider the
induced map f̂ : P̂ −→ M . Suppose that the pull-back bundle f̂ ∗(TM) −→ P̂ is
trivial. Then f is homotopic through 〈k〉-maps to a 〈k〉-immersion.
Proof. Since both T P̂ → P̂ and f̂ ∗(TM) → P̂ are trivial vector bundles and
dim(M) > dim(P ), we may choose a bundle injection T P̂ → f̂ ∗(TM) covering the
identity on P̂ , and hence a fibrewise injective bundle map ψ̂ : T P̂ −→ TM that
covers the map f̂ . Using the quotient construction from (11.7), the bundle map ψ̂
induces a unique formal 〈k〉-immersion ψ ∈ Immf〈k〉(P,M) whose underlying 〈k〉-
map is f . It then follows from Theorem 11.2.3 that there exists a 〈k〉-immersion
φ ∈ Imm〈k〉(P,M) such that D(φ) is on the same path component as ψ. It then
follows that φ is homotopic through 〈k〉-maps to the map that underlies ψ, which is
f . This completes the proof of the corollary.
11.4. The Self-Intersections of a 〈k〉-Immersion
For what follows let M be a manifold of dimension m and let P be a 〈k〉-
manifold of dimension p. We will need to analyze the self-intersections of 〈k〉-
immersions P →M .
Definition 11.4.1. For M a manifold and P a 〈k〉-manifold, a 〈k〉-immersion f :
P −→M is said to be in general position if the following conditions are met:
i. The immersion fβ : βP →M is self-transverse.
ii. The restriction map f |Int(P ) : Int(P ) −→ M is a self-transverse immersion and
is transverse to the immersed submanifold fβ(βP ) ⊂M .
119
Let f : P −→ M be a 〈k〉-immersion that is in general position. Let qˆ : P −→ P̂
denote the quotient projection and let 4̂P ⊂ P × P be the subspace defined by
setting
4̂P = (qˆ × qˆ)−1(4P̂ ),
where 4P̂ ⊂ P̂ × P̂ is the diagonal subspace. It follows from Definition 11.4.1 that
the map
(f × f)|(P×P )\4̂P : (P × P ) \ 4̂P −→M ×M
is transverse to the diagonal submanifold 4M ⊂ M × M . We denote by Σf ⊂
(P × P ) \ 4̂P the submanifold given by
Σf :=
(
(f × f)|(P×P )\4̂P
)−1
(4M). (11.8)
By the techniques of Section 7.3, Σf has the structure of a 〈k, k〉-manifold with
∂1Σf = f |∂1P t f, ∂2Σf = f t f |∂1P , ∂1,2Σf = f |∂1P t f |∂1P ,
β1Σf = fβ t f, β2Σf = f t fβ, β1,2Σf = fβ t fβ.
The involution
P × P \ 4̂P −→ P × P \ 4̂P , (x, y) 7→ (y, x)
restricts to an involution on Σf ⊂ P × P \ 4̂P which we denote by
TΣf : Σf −→ Σf . (11.9)
120
It is clear that the involution TΣf has no fixed-points. Since
∂1Σf ⊂ (∂1P )× P and ∂2Σf ⊂ P × (∂1P ),
it follows that
TΣf (∂1Σf ) ⊂ ∂2Σf and TΣf (∂2Σf ) ⊂ ∂1Σf .
If both M and P are oriented, then Σf obtains a unique orientation induced
from orientations on P and M in the standard way. Furthermore, TΣf preserves
orientation if m− p is even and reverses orientation if m− p is odd. We sum up the
observations made above into the following proposition.
Proposition 11.4.1. Let P be an oriented 〈k〉-manifold of dimension p and let M
be an oriented manifold of dimension m. Let f : P −→ M be a 〈k〉-immersion
which is in general position. Then the double-point set Σf has the structure of an
oriented 〈k, k〉-manifold of dimension 2p −m, equipped with a free involution TΣf :
Σf −→ Σf such that
TΣf (∂1Σf ) ⊂ ∂2Σf and TΣf (∂2Σf ) ⊂ ∂1Σf .
The involution TΣf preserves orientation if m− p is even and reverses orientation if
m− p is odd.
11.5. Modifying Self-Intersections
In this section, we develop a technique for eliminating the self-intersections
of a 〈k〉-immersion P → M by deforming the 〈k〉-immersion to a 〈k〉-embedding
via a homotopy through 〈k〉-maps. We will solve this problem in the special case
121
that P is a 2-connected, oriented, (2n + 1)-dimensional 〈k〉-manifold and M is a
2-connected, oriented, (4n+ 1)-dimensional manifold and n ≥ 2.
By Proposition 11.4.1, if f : P −→ M is such a 〈k〉-immersion in general
position, then the double-point set Σf is a 1-dimensional 〈k, k〉-manifold with an
orientation preserving, involution T : Σf −→ Σf with no fixed points, such that
T (∂1Σf ) = ∂2Σf and T (∂2Σf ) = ∂1Σf .
We will need the following general result about such 1-dimensional, 〈k, k〉-manifolds
equipped with such an involution as above.
Lemma 11.5.1. Let N be a 1-dimensional, closed, oriented, 〈k, k〉-manifold.
Suppose that N is equipped with an orientation preserving, involution T : N −→ N
with no fixed points, such that
T (∂1N) = ∂2N and T (∂2N) = ∂1N.
Then,
β1N = β2N = +〈j〉 unionsq −〈j〉
for some integer j.
Proof. We prove this by contradiction. Suppose that β1N = +〈j〉 unionsq −〈l〉 where
j 6= l. Since T preserves orientation and T (∂1N) = ∂2N and T (∂2N) = ∂1N , it
follows that β2N = +〈j〉 unionsq −〈l〉 as well.
122
If we forget the 〈k, k〉-structure on N , then N is just an oriented, 1-
dimensional manifold with boundary equal to
∂1N unionsq ∂2N = [(+〈j〉 unionsq −〈l〉)× 〈k〉]
⋃
[(+〈j〉 unionsq −〈l〉)× 〈k〉]. (11.10)
By reorganizing the above union, we see that the zero-dimensional manifold in
(11.10) is equal to +〈2 · k · j〉 unionsq −〈2 · k · l〉.
However since j 6= l, there is no oriented, one dimensional manifold with
boundary equal to +〈2 · k · j〉 unionsq −〈2 · k · l〉. This yields a contradiction. This proves
the lemma.
Proposition 11.5.2. Let P be a closed 〈k〉-manifold of dimension 2n + 1, let M
be a manifold of dimension 4n + 1 and let f : P −→ M be a 〈k〉-immersion.
Then there is a regular homotopy (through 〈k〉-immersions) of f to a 〈k〉-immersion
f ′ : P −→M , such that
β1Σf ′ = β2Σf ′ = f
′
β(βP ) ∩ f ′(Int(P )) = ∅.
Proof. First, by choosing a small, regular homotopy, we may assume that f is in
general position. Since βP is a closed 2n-dimensional manifold and 2n < 4n+1
2
, the
fact that f is in general position implies that fβ : βP −→M is an embedding.
Furthermore, we may assume that fβ(βP ) is disjoint from the image of the
double point set of the immersion f |Int(P ) : Int(P ) −→M .
Consider the intersection fβ(βP ) ∩ f(Int(P )). We choose a closed, disk
neighborhood U ⊂ Int(P ) that contains f |−1Int(P )(fβ(βP )), such that the restriction
f |U : U −→ M is an embedding (we may choose U so that f |U is an embedding
because fβ(βP ) is disjoint from the image of the double point set of f |Int(P )). By
123
Lemma 11.5.1 it follows that there is a diffeomorphism
f |−1U (fβ(βP )) ∼= β1Σf ∼= +〈j〉 unionsq −〈j〉
for some integer j, and so, the oriented, algebraic intersection number associated to
the intersection f(U) ∩ fβ(βP ) is equal to zero. By the Whitney trick, we may find
an isotopy through embeddings φt : U −→M with
φ0 = f |U and φt|∂U = f |∂U for all t ∈ [0, 1]
such that φ1(U) ∩ fβ(βP ).
We then may extend this isotopy over the rest of P by setting it equal to f
for all t ∈ [0, 1] on the compliment of U ⊂ P . This concludes the proof of the
lemma.
Corollary 11.5.3. Let P be a 2-connected, closed, oriented 〈k〉-manifold of
dimension 2n + 1. Let M be a 2-connected, oriented, manifold of dimension 4n + 1,
and let f : P −→ M be a 〈k〉-immersion. Then f is homotopic through 〈k〉-maps to
a 〈k〉-embedding.
Remark 11.5.1. In the statement of the above corollary, we are not asserting that
any 〈k〉-immersion f : P −→ M is regularly homotopic to a 〈k〉-embedding. The
homotopy through 〈k〉-maps constructed in the proof of this result may very well
not be a homotopy through 〈k〉-immersions.
Proof. Assume that f is in general position. By the previous proposition we may
assume that fβ : βP −→ M is an embedding and that β1Σf = ∅. We may choose a
124
collar embedding
h : ∂1P × [0,∞) −→ P with h−1(∂1P ) = ∂P1 × {0},
such that for each i ∈ 〈k〉, the restriction map
f |h(∂i1P×[0,∞)) : h(∂i1P × [0,∞)) −→M
is an embedding, where ∂i1P = Φ
−1(βP × {i}). Now let U ⊂ M be a closed tubular
neighborhood of fβP (βP ) ⊂M , disjoint from the image f(P \h(∂1P × [0,∞))), such
that the boundary ∂U is transverse to f(P ). We define,
Z := M \ Int(U), P ′ := f−1(Z), f ′ := f |P ′ . (11.11)
By construction, P ′ and Z are a manifolds with boundary, f ′ maps ∂P ′ into ∂Z,
and f ′(Int(P ′)) ⊂ Int(Z). The corollary will be proven if we can find a homotopy of
f ′, relative ∂P ′, to a map
f ′′ : (P ′, ∂P ′) −→ (Z, ∂Z)
which is an embedding. Using the 2-connectivity of both P ′ and Z (and the
dimensional conditions on P ′ and Z), the existence of such a homotopy follows
from (12, Theorem 4.1) (or from (19, Theorem 1.1)).
125
11.6. Representing 〈k〉-Maps by 〈k〉-Embeddings
We are now in a position to prove Theorem 5.4.1 from Section 5.4. It follows
as a corollary of the results developed throughout this section. Here is the theorem
restated again for the convenience of the reader.
Theorem 11.6.1. Let n ≥ 2 be an integer and let k > 2 be an odd integer. Let
M be a 2-connected, oriented manifold of dimension 4n + 1. Then any 〈k〉-map
f : V 2n+1k −→M is homotopic through 〈k〉-maps to a 〈k〉-embedding.
Proof. Since M is 2-connected, it follows that the map f̂ : V̂ 2n+1k −→ M (which
is the map induced by the 〈k〉-map f), extends to a map M(Z/k, 2n) −→ M ,
where M(Z/k, 2n) is a Z/k-Moore-space (see Lemma 5.3.1). It then follows that
the vector bundle f̂ ∗(TM) −→ V̂ 2n+1k is classified by a map V̂ 2n+1k −→ BSO that
factors through a map M(Z/k, 2n) −→ BSO. When k is odd, the Z/k-homotopy
group pi2n(BSO;Z/k) is trivial. It follows that the bundle f̂ ∗(TM) −→ P̂ is
trivial. Now, it is easy to verify that the 〈k〉-manifold V 2n+1k is parallelizable as
a 〈k〉-manifold (see Definition 11.3.1). It then follows from Corollary 11.3.1 that
the map f is homotopic through k-maps to a 〈k〉-immersion, which we denote by
f ′ : V 2n+1k −→ M . The proof of the theorem then follows by applying Corollary
11.5.3 to the 〈k〉-immersion f ′.
126
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