SCALAR CURVATURE AND TRANSFER MAPS IN Spin AND Spinc BORDISM
by
ELLIOT GRANATH
A DISSERTATION
Presented to the Department of Mathematics
and the Division of Graduate Studies of the University of Oregon
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
June 2023
DISSERTATION APPROVAL PAGE
Student: Elliot Granath
Title: Scalar Curvature and Transfer Maps in Spin and Spinc Bordism
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:
Boris Botvinnik Chairperson
Nicolas Addington Core Member
Robert Lipshitz Core Member
Peng Lu Core Member
Spencer Chang Institutional Representative
and
Krista Chronister Vice Provost for Graduate Studies
Original approval signatures are on file with the University of Oregon Division of
Graduate Studies.
Degree awarded June 2023.
2
© 2023 Elliot Granath
This work is licensed under a Creative Commons
Attribution License.
3
DISSERTATION ABSTRACT
Elliot Granath
Doctor of Philosophy
Department of Mathematics
June 2023
Title: Scalar Curvature and Transfer Maps in Spin and Spinc Bordism
In 1992, Stolz proved that, among simply connected Spin-manifolds of dimension
5 or greater, the vanishing of a particular invariant α is necessary and sufficient
for the existence of a metric of positive scalar curvature. More precisely, there is
a map α : ΩSpin∗ → ko (which may be realized as the index of a Dirac operator)
which Hitchin established vanishes on bordism classes containing a manifold with
a metric of positive scalar curvature. Stolz showed kerα is the image of a transfer
map ΩSpinBPSp(3) → ΩSpin∗−8 ∗ . In this paper we prove an analogous result for Spinc-
manifolds and a related invariant cαc : ΩSpin∗ → ku. We show that kerαc is the
sum of the image of Stolz’s transfer ΩSpin c∗−8BPSp(3) → ΩSpin∗ and an analogous map
SpincΩ Spin
c
∗−4 BSU(3) → Ω∗ . Finally, we expand on some details in Stolz’s original paper
and provide alternate proofs for some parts.
4
ACKNOWLEDGMENTS
Thank you to my advisor Boris Botvinnik and to my family, including mom, dad,
Wes, Ashley, Reggie, Marge, Barbara, and Pancake.
5
For Grandma Marge
6
TABLE OF CONTENTS
Chapter Page
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.1. Twisted scalar curvature for Spinc manifolds . . . . . . . . 14
1.2. Existence of positive L-twisted scalar curvature . . . . . 15
1.3. New transfer map and main result . . . . . . . . . . . . . . . 16
1.4. Proof of main result (theorem 1.7) . . . . . . . . . . . . . . . 17
2. ALGEBRAIC BACKGROUND . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1. The Steenrod algebra and its dual . . . . . . . . . . . . . . . 26
2.2. A closer look at A∗ . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3. Pairing between A and A∗ . . . . . . . . . . . . . . . . . . . . . 32
2.4. The Hopf algebra antipode map and Hopf subalgebras . . 33
2.5. A brief review of comodule theory . . . . . . . . . . . . . . . 35
2.6. A(1) modules and H∗BPSp(3) . . . . . . . . . . . . . . . . . . . . 36
2.7. E(1) modules and H∗BSU(3) . . . . . . . . . . . . . . . . . . . . . 39
2.8. ko and ku . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.9. Extended modules and indecomposable quotients . . . . . . 44
3. TRANSFER MAPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1. Virtual vector bundles . . . . . . . . . . . . . . . . . . . . . . . 46
3.2. Thom spectra and Dold’s theorem . . . . . . . . . . . . . . . . 48
3.3. The general bundle transfer map . . . . . . . . . . . . . . . . 50
Virtual bundles and generalized Thom isomorphisms . . . . . . . . . . 50
The Thom map associated to a bundle . . . . . . . . . . . . . . . . . 52
Constructing the umkehr map from the Thom map . . . . . . . . . . . 53
The HP2-bundle transfer . . . . . . . . . . . . . . . . . . . . . . . . . 55
7
3.4. The CP2-bundle transfer . . . . . . . . . . . . . . . . . . . . . . 57
APPENDICES
A. THE A(1)-ACTION ON H∗BPSp(3) . . . . . . . . . . . . . . . . . . . . . 60
A.1. Action by A(1) on BPSp(3) . . . . . . . . . . . . . . . . . . . . . . 65
A.2. Cohomology of BPSp(2, 1) . . . . . . . . . . . . . . . . . . . . . . 68
B. PRIMITIVE GENERATORS . . . . . . . . . . . . . . . . . . . . . . . . . 70
C. COMPUTATION OF THE Spinc TRANSFER MAP . . . . . . . . . . . . 75
REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
8
CHAPTER 1
INTRODUCTION
A major topic of differential geometry today is the study of Riemannian curvatures
on a smooth manifold, and how this relates to topological invariants. Perhaps the
simplest notion of curvature on a Riemannian manifold (M, g) is the scalar curvature
sg. One way to define curvature is by comparing the volume of a geodesic ball Bε
around a point x ∈ M with a ball (0)Bε of radius ε in Euclidean space. More precisely,
sg(x) is such that ( )
Vol(Bε) − sg(x)= 1 ε2 +O(ε4) .
(0)
Vol(Bε ) 6(n+ 2)
In this document we address a fundamental question about Riemannian metrics
of positive scalar curvature (psc metrics).
Under which conditions does a manifold admit a psc metric? (1.1)
The case for 2-dimensional manifolds is unique: here the scalar curvature is simply
twice the Gauss curvature Kg.∫The Gauss-Bonnet theorem states
sg(x)dσg = 4πχ(M),
M
where σg is the volume element corresponding to g and χ(M) is the Euler characteris-
tic of M . Hence, in 2 dimensions, the existence of metrics of positive scalar curvature
is a matter of which surfaces have positive Euler characteristic – namely, S2 and RP2.
The existence problem 1.1 has also been resolved for all manifolds of dimension
five or greater which are simply connected. Under these conditions, Gromov and
Lawson proved a crucial result of surgery theory. We recall some core definitions and
facts of this field. By default, all manifolds we mention in this paper are compact.
A surgery on a manifold is defined as follows: let Sp×Dq+1 → M be an embedding
of a sphere Sp with a tubular neighborhood (a “handle”) around it, where p+ q+1 is
9
the dimension of M . The surgery on M given by this embedding produces a manifold
M ′ obtained by removing the interior of Sp × Dp+1 ⊂ M and gluing in the handle
Dp+1 × Sq:
′ ( ( )) ( )M = M \ int Sp ×Dq+1 ∪ Dp+qSp×Sq × Sq .
In short, the effect of this surgery is to “suture up” the p-dimensional hole given by a
topologically framed embedding of Sp. We say that the codimension of this surgery
is q + 1, since this is the codimension of Sp in M . Gromov and Lawson established
an important result about Spin manifolds and metrics of positive scalar curvature:
Theorem 1.1 ([GL]). Let (M, g) be a compact Riemannian manifold with positive
scalar curvature (a psc metric). If M ′ is obtained from M by a surgery of codimension
≥ 3, then M ′ carries a psc metric.
Theorem 1.1 has the following important corollary and related result which, for
some classes of manifolds, allows us to only consider bordism classes when discussing
the existence of psc metrics. For a review of X-bordism, see [Switz, §12.12].
Corollary 1.2 ([GL]). Let M be a simply connected Spin manifold of dimension
≥ 5. If M is Spin-bordant to a manifold M ′ with a psc metric, then M also has a
psc metric.
For manifolds which are not Spin, Gromov and Lawson proved the following.
Theorem 1.3 ([GL]). Let M be a simply connected manifold of dimension ≥ 5 which
is not Spin. Then M admits a psc metric.
Remark. There is inconsistency in the literature on the usage of “bordism” versus
“cobordism.” Following Rudyak ([Rudyak, 7.19]), we tend to use “bordism” ring and
say that two manifolds are “bordant.” This terminology may be more common today
10
since it allows authors to reserve “cobordism” for the dual cohomology theory (as in
[Rudyak, 7.30] and [Switz, §12]). Historically the term “cobordism” was more common
for the homology theory (as in [Stong]). This is presumably since “cobordant” has a
literal interpretation of two manifolds which together bound another manifold (the
French verb “bordant” means “to bound”).
We can summarize corollary 1.2 and theorem 1.3 and rephrase them as follows:
for M a simply connected manifold of dimension n ≥ 5,
• if M is Spin, then the existence of a psc metric on M depends only on the Spin
bordism class [M ] ∈ ΩSpinn ;
• if M is not Spin, then the existence of a psc metric on M depends only on the
oriented bordism class [M ] ∈ ΩSOn .
Remark. In particular, for simply connected manifolds of dimension ≥ 5, the existence
of a psc metric is determined by purely topological data. Similarly, a manifold M
is Spin if and only if w2(M) = 0 and simply connected if and only if w1(M) = 0
(assuming M is connected). The equivalence classes of Spin-structures on a manifold
are in bijection with H1(M) (with coefficients in Z2), hence, if a simply connected
manifold admits a Spin-structure, the Spin-structure is unique.
Since at least the early 1960s, it was known that the presence of a Spin-structure on
a manifold was had an intimate relationship with that of a psc metric. A Riemannian
manifold (M, g) which is Spin has a Dirac operator Dg acting on the bundle of spinors
over M , and in 1963 Lichnerowicz ([Lich]) proved the formula
1
D2 ∗g = ∇ ∇+ sg, (1.2)4
where ∇ is a covariant derivative and ∇∗ is the formal adjoint. Equation (1.2) implies
that if sg is positive, Dg is invertible (i.e., there are no harmonic spinors on M). If
11
M is a Spin manifold of dimension n with a psc metric, the Lichnerowicz formula
implies that [M ] ∈ ΩSpinn lies in the kernel of the index
α : ΩSpinn → kon . (1.3)
Following Stolz’s work, we had a complete answer to the existence problem 1.1 for
simply connected manifolds of dimension ≥ 5. Under these conditions,
(i) if M is not Spin, then M admits a psc metric ([GL]);
(ii) if M is Spin, then M has a psc metric if and only if α([M ]) = 0 ([Stolz]).
Remark. In [GL], Gromov and Lawson proved the rational version of Stolz’s result,
which can be stated as follows: for a simply connected Spin-manifold of dimension
n ≥ 5, if α([M ]) = 0, then some multiple M#M# · · ·#M admits a psc metric.
The techniques used by Stolz in his proof are central to the results in this paper.
We now describe the geometric idea behind Stolz work. Let HP2 be the quaternionic
projective space with the standard metric g0. It is well known that the group of
isometries of HP2 (with respect to g0) is the projective group PSp(3) (which is the
quotient of Sp(3) by {±1}). We now make an elementary observation.
Lemma 1.4. Let E be the total space of a smooth bundle over a manifold M with
fiber HP2 and structure group PSp(3). Then E carries a psc metric.
Proof. Cover M by open sets U1, . . . , Un over which E is locally (diffeomorphic to)
Di×HP2. Let gM be the the metric onM and g0 be the standard metric onHP2. Since
M is compact, sg bounded, whence there exists sufficiently small λ > 0 such thatM
g := gM ×λg0 has positive scalar curvature on each neighborhood: sg = s −1g +λ sM g0 .
Again for λ sufficiently small, this holds over each of the Ui. Finally, since PSp(3)
acts isometrically on HP2, these local metrics glue to form a global psc metric.
12
Remark. It is an elementary fact that, for any λ > 0, the scaled metric λg is a
Riemannian metric with scalar curvature sλg = λ−1sg. In fact, since scalar curvature
can be defined as a sum of sectional curvature, one’s geometric intuition for surfaces
(roughly) shows this for higher dimensions.
Of course PSp(3) is a compact Lie group, so we have a classifying space BPSp(3)
with a universal PSp(3)-bundle and the associated HP2-bundle. We now define a
transfer map
Ψ: ΩSpinn BPSp(3) → Ω
Spin
n+8 (1.4)
as follows: a class in ΩSpinn BPSp(3) is represented by a pair (M, f), where M is an
n-dimensional Spin manifold and f is a map M → BPSp(3). Let M̂ be the total
space of the pullback via f of the HP2-bundle. Note that M̂ → M is an HP2-bundle
which also has the structure group PSp(3) acting isometrically on the fibers. Now we
have the following main result from [Stolz].
Theorem 1.5 (Stolz). For the transfer map Ψ: ΩSpinBPSp(3) → ΩSpinn n+8 and index
map α : ΩSpinn → kon, we have imΨ = kerα.
Since Lichnerowicz’s formula implies imΨ ⊆ kerα, Stolz had to show kerα ⊆
imΨ. He achieved this by reducing to a homotopy-theoretic problem: to begin with,
the transfer Ψ may be considered as the map induced on homotopy of a map of
spectra
T : MSpin∧Σ8BPSp(3)+ → MSpin . (1.5)
The index map α is similarly induced by a map
D : MSpin → ko . (1.6)
The algebraic fact that α([M ]) = 0 if M has a psc metric, for example, translates
to statement that DT is nullhomotopic. This means that T factors through the
13
homotopy fiber M̂Spin of D. We write T̂ for this lift and i : M̂Spin → MSpin for the
inclusion of M̂Spin. Theorem 1.5 then follows from showing T̂ induces a surjection of
homotopy groups.
It is worth emphasizing here that, prior to Stolz’s result in [Stolz], the same
result had been proven rationally in [GL]. As was well known by that time, the map
MSpin → MSO is a homotopy equivalence for any coefficient ring containing 1/2.
Hence the most difficult component is the 2-primary data. Hence, in this paper, we
work entirely with coefficients in Z2 = Z/(2) for homology and cohomology. We may
also abuse notation and write π∗X to mean π∗(X)⊗ Z(2).
1.1 Twisted scalar curvature for Spinc manifolds
As we have seen, existence of a Spin structure (and consequently of the Dirac
operator) plays a fundamental role in the existence of a psc metric. More generally,
a Spinc-structure implies the existence of a Spinc Dirac operator.
Let M be a simply-connected manifold with w2(M) ̸= 0 (i.e. M is not a spin
manifold). Then M has a Spinc-structure if there is a class c ∈ H2(M ;Z) which
maps to w2(M) under the mod-2 reduction map H2(M ;Z) → H2(M ;Z2). The class
c gives a map c : M → CP∞ and, consequently, a complex line bundle L → M . We
use the notation (M,L) for a manifold with a choice of Spinc-structure. We notice
that a Spin-manifold M has a canonical Spinc corresponding to the trivial complex
line bundle over M .
Assume (M,L) is a non-Spin Spinc manifold. We choose a Riemannian metric g
on M , a Hermitian metric h on L, and a unitary connection AL on the line bundle
14
L. These data give the Spinc Dirac operator D(M,L). Then we have the Lichnerowicz
formula [Lich]
D2 ∗
1
(M,L) = ∇∇ + sg +RL (1.7)4
here
1 ∑RL = FL(ej, ek) · ej · ek,
2
j<k
where one sums over an orthonormal frame, and where FL is the curvature of the
connection AL on the line bundle L. We denote
sLg := sg + 4RL,
and say that sLg is the L-twisted scalar curvature. Notice that sLg is, in fact, a zeroth
order operator on spinors, and it depends on a choice of the hermitian metric h on L
and on the connection AL.
1.2 Existence of positive L-twisted scalar curvature
One of our primary goals is to resolve the following existence question: under
which conditions on a non-Spin Spinc manifold (M,L) does there exist a Riemannian
metric g, a Hermitian metric h on L, and a connection AL such that the L-twisted
scalar curvature sLg is positive?
It turns out that the positivity of the L-twisted scalar curvature sLg is also invariant
under surgeries of codimension at least three. In particular, this implies that for a
simply-connected manifold M the existence of the data (g, h, AL) as above such that
sL is a positive operator, depends only on the Spinc-cobordism class [(M,L)] ∈ ΩSpincg .
Next, we recall that the index of the Spinc Dirac operator D(M,L) takes values
in ku∗, where ku is the connective cover of the complex K-theory spectrum KU. In
15
particular, we have a Spinc-version of the α invariant
αc : ΩSpin
c
n → kun . (1.8)
The formula (1.7) shows that, if the L-twisted scalar curvature sLg is positive, then
the cSpinc-bordism class [(M,L)] ∈ ΩSpin is in the kernel of the index map αcn . This
leads to the following existence result due to Botvinnik and Rosenberg.
Theorem 1.6 ([BR22; BR18]). Let (M,L) be a simply connected non-Spin Spinc
manifold of dimension n ≥ 5. Then M admits a Riemannian psc metric g, a her-
mitian metric h and a connection AL such that sLg > 0 if and only if the Spin
c index
αc[(M,L)] vanishes in kun.
A proof of Theorem 1.6 is given in [BR22] Theorem 3.8 and [BR18] Corollary 32
which relies on deep results in cobordism and homotopy theory. A main goal of this
thesis is to give a direct proof of Theorem 1.6 using the same technology developed
by Stolz in [Stolz].
As with the Spin case, we will ignore odd-primary data. The proof for the case of
odd primes also appears in [BR22]; in this paper we give a direct, constructive proof
for the 2 primary case.
1.3 New transfer map and main result
We notice that the complex projective spaces CP2k are non-spin manifolds. In par-
ticular, CP2 has a standard Spinc-structure given by a complex line bundle L0 → CP2
with c1(L0) = x, where x is a generator ofH2(CP2;Z). A calculation of αc[(CP2k, L0)]
was worked out by Hattori [Hat], in particular, he shows that αc[(CP2, L0)] = 0.
The projective plane CP2 has a remarkable property – namely, the group G =
SU(3) acts transitively on CP2, and CP2 = G/H, where the subgroup H := SU(2, 1)
16
is the subgroup of elements in U(2) × U(1) ⊂ U(3) with determinant 1. We obtain
a fiber bundle p : BH → BG with fiber CP2 and structure group SU(3). Thus
given a Spinc-manifold (M,L) and a map f : M → BG, we can form the associated
CP2-bundle p̂ : E → M as a pull-back
E BH
p̂ p
f
M BG
where E = M ×f CP2 has dimension n+ 4 and has a Spinc structure inherited from
the Spinc structure on M defined by L and the Spinc structure on CP2 defined by
the bundle L. This construction defines a transfer map
c
T c : ΩSpin Spincn−4 (BSU(3)) → Ωn . (1.9)
We also have a Spinc-version of the Stolz transfer map:
T : ΩSpin
c Spin
n−8 (BPSp(3)) → Ωn . (1.10)
Here is the main result of this thesis:
Theorem 1.7. The transfer maps T and T c are such that
im(T ) + im(T c) = kerαc
as abelian groups.
1.4 Proof of main result (theorem 1.7)
We summarize and expand on Stolz’s original proof and give an analogous result
for Spinc
c
bordism involving the transfer map T c : ΩSpin BSU(3) → ΩSpinc∗−4 ∗ of (1.9).
The Steenrod algebra A and basic theory of modules and comodules over A (and
17
its dual) are needed to describe the transfer maps, so we begin with some algebraic
preliminaries in chapter 2. In chapter 3 we give more details about the transfer maps
T and T c and compute their induced maps on cohomology. Important lemmas and
other results used in the proof of theorem 1.7 appear in chapter 3, with some necessary
computations included in the appendices.
The compact Lie group PSp(3) acts transitively on S11 ⊂ H3 and this descends to a
transitive action on HP2. The stabilizer of [0 : 0 : 1] is PSp(2, 1) := P(Sp(2)×Sp(1)),
giving a fiber bundle PSp(2, 1) → PSp(3) → HP2. In turn this yields a bundle
π : BPSp(2, 1) → BPSp(3) with fiber HP2 and structure group PSp(3). Note that
the standard metric on HP2 has positive scalar curvature and the structure group
PSp(3) acts via isometries with respect to the standard metric.
Using observations in the last paragraph we obtain the transfer map of Stolz
T : ΩSpinn−8BPSp(3) → ΩSpinn as follows: a class in ΩSpinn BPSp(3) can be written [M, f ],
where M is an n-dimensional Spin manifold and f is a map M → BPSp(3). Let
M̂ be the total space of the pullback via f of our HP2-bundle. Now [M̂ ] is a class
in ΩSpinn+8 , and the transfer map T takes [M, f ] to [M̂ ]. Now the natural inclusion
MSpin → MSpinc lets us easily translate this to Spinc-bordism, although we abuse
c
notation slightly and reuse the map names, e.g. T c: ΩSpinn−8 BPSp(3) → ΩSpinn . This
notation is different from that of Stolz, who writes Ψ instead of T .
Analogously to the quaternionic case, the special unitary group SU(3) acts tran-
sitively on S5 ⊂ C3 yielding a transitive action on CP2. As before, the stabi-
lizer of [0 : 0 : 1] is SU(2, 1) := S(U(2) × U(1)) and we hence obtain a bundle
BSU(2, 1) → BSU(3) with fiber CP2. This bundle admits a Spinc structure, giving a
map BSU(3) → BSpinc. In this case we get a transfer map T : ΩSpin
c
n−4 BSU(3) → ΩSpin
c
n
with the property imT c ⊆ kerαc. Unfortunately, the analogy with the Spin case
ends here, as the reverse containment fails. Moreover, we will see that neither T
18
nor T c are surjective onto kerαc; instead we combine the two transfers and show
im T + imT c = kerαc.
Via the classical Pontrjagin-Thom construction ΩSpincn (X) can be identified with
πn(MSpin
c ∧X+). The transfer map T is the map on homotopy groups induced by a
map of spectra T : MSpinc∧Σ8BPSp(3)+ → MSpinc; similarly T c is induced by a map
T c : MSpinc∧Σ4BSU(3) → MSpinc. The αc invariant corresponds to a spectrum map
Dc : MSpinc → ku, and we have the following diagram of maps, where i : M̂Spinc →
MSpinc is inclusion of the homotopy fiber of Dc and µ : MSpinc ∧MSpinc → MSpinc
is the ring spectrum product map.
M̂Spinc
T̂∨T̂ c
i
MSpinc ∧ (Σ8BPSp(3)+ ∨
µ
Σ4BSU(3)+) MSpin
c ∧MSpinc MSpinc
Dc
T∨T c
ku
The unlabeled map is 1∧ (t∨ tc), where t and tc are the respective bundle transfer
maps composed with the Thom map. That is,
• t : Σ8BPSp(3)+ → MSpinc is the composition t = M◦ T ;
• T : Σ8BPSp(3)+ → M(−τ) is the bundle transfer map of [Board, §VI.6],
• where −τ is the stable complement of the bundle of tangent vectors along
the fibers of BPSp(2, 1) → BPSp(3);
• M : M(−τ) → MSpinc is the Thom map induced by the classifying map of −τ .
Our main result can now be restated as follows:
19
Theorem 1.8. On homotopy groups, T ∨ T c induces a surjection
( )
ΩSpin
c
Σ8∗ BPSp(3)+ ∨ Σ4BSU(3)+ → π∗M̂Spinc.
Remark 1.9. After localizing at 2, the spectrum MSpinc splits into a wedge of sus-
pensions of ku with other spectra. The map Dc is simply projection onto the lowest
degree copy of ku. Details can be found in [HH, §5]. The analogous statements hold
for MSpin and ko. These facts give a particularly nice description of M̂Spinc and
M̂Spin which make these homotopy fibers reasonable objects to work with.
To prove theorem 1.8 (at the prime 2), we use the Adams spectral sequence along
with homological data. More precisely, we will show that T̂ ∨ T̂ c induces a split
surjection of A∗-comodules, hence a surjection of the E2-terms
Exts,tA (Z2, H∗MSpin
c ⊗ (H∗−8BPSp(3)⊕H s,t∗−4BSU(3))) → ExtA (Z2, H∗M̂Spin
c)
∗ ∗
(1.11)
which converge to
π∗(MSpin
c ∧ (Σ8BPSp(3) 4 c+ ∨ Σ BSU(3)+))⊗ Z(2) → π∗(M̂Spin )⊗ Z(2) (1.12)
It turns out that the first of these spectral sequences has no nontrivial differentials,
hence the surjection (1.11) at the level of E2 pages implies the surjection (1.12) on
the E∞ pages.
To prove theorem 1.7, then, it remains to show T̂∗ ∨ T̂ c∗ is a split surjection of
A∗-comodules. Several algebraic maneuvers make our task easier: a useful change-of-
rings construction reduces our task to working with comodules over a sub-coalgebra
E(1)∗ ⊂ A∗. We also prefer to dualize to cohomology in order to work withA-modules
and E(1)-modules rather than comodules, although most results are stated in terms
of comodules where possible. Finally, we give an algebraic result in Theorem 2.14
20
which provides an easy way to show that an injection of E(1)-modules (or a surjection
of E(1)∗-comodules) is split. Following [Pete], we also know
H∗BSpin
c = Z2[x2i , xj : α(i) < 3, α(j) ≥ 3],
where α(n) is the number of nonzero terms in the base-2 expansion of n.
Theorem 1.10. We have a commuting diagram as shown, where ρ : BSpinc → BO
is the projection and D, U and ι are orientation classes. We make some observations
about this diagram. We have the standard identification H∗HZ2 = Z2[ζ1, ζ2, . . .], and,
MSpinc D ku
Mρ ι
MO U HZ2
via the Thom isomorphism, we can write H∗MO = Z2[x1, x2, . . .], where deg(xi) = i.
The spectrum MO is HZ2-oriented via a Thom class (i.e., an orientation) U : MO →
HZ2. Similarly Dc : MSpinc → ku can be considered as the ku-orientation of MSpinc.
We use the Thom isomorphism to identify H∗MO with H∗BO = Z2[x1, x2, x3, . . .],
where deg(xi) = i. The dual Steenrod algebra A∗ is the polynomial ring Z2[ξ1, ξ2, . . .],
although we prefer to use the Hopf conjugate generators and write A∗ = Z2[ζ1, ζ2, . . .]
(here deg(ξi) = deg(ζi) = 2i − 1). Of course A∗ = H∗HZ2, and the inclusion ko →
HZ2 induces on homology the injection
Z2[ζ21 , ζ22 , ζ3, ζ4, . . .] → Z2[ζ1, ζ2, ζ3, ζ4, . . .].
1. The maps Mρ and ι induce monomorphisms on homology.
2. We can identify H∗MSpinc = im(Mρ ) = Z [x2∗ 2 i , xj : α(i) < 3, α(j) ≥ 3].
3. Define P ⊂ H∗MO as P = Z2[x2i−1 : i ≥ 1]. Then U∗ maps P isomorphically
onto H∗HZ2.
21
4. Identifying P ⊂ H∗MSpinc, let
R = P ∩H∗MSpinc = Z2[x21, x23, x7, x15, . . .] ⊂ H∗MSpinc.
Then D∗ maps R isomorphically onto H∗ ku.
Proof. The first claim is immediate from the fact that the corresponding maps on
cohomology are quotients (see [Stong], for example). The computation of H∗MSpinc
and H∗MSpinc can be found in [Switz]. The third claim is [Stolz, Corollary 4.7], and
the last follows from the nontrivial fact that D admits a splitting H ku → H MSpinc∗ ∗ ∗
([Hat], [Stolz2]).
There is a useful functorial construction for MSpinc-module spectra which identi-
fies H∗X with A∗□E(1)∗H∗X, where H∗X := Z2⊗RH∗X is a subalgebra of H∗MSpinc
which maps isomorphically to H∗ ku via Dc∗. The algebra E(1) is generated by
Q 10 = Sq and Q = Sq1 Sq2+Sq21 Sq1; the cotensor product □ is also standard nota-
tion and also defined in chapter 2. (Note that R = A∗□E(1)∗Z2.) This construction,
which is detailed in theorem 2.15, is functorial in the sense that maps f : H∗X → H∗Y
induce maps f : H∗X → H∗Y . This method allows us to work with modules over
E(1) (or comodules over E(1)∗) rather than using the whole Steenrod algebra.
Applying this construction to the diagram which appeared just before theorem
1.8 yields the following diagram.
22
H MSpinc∗ ⊗ (H∗Σ8BPSp(3) 4+ ⊕H∗Σ BSU(3)+)
1⊗(t∗⊕tc∗)
H MSpinc c∗ ⊗H∗MSpin
µ
H MSpinc∗
D∗
H∗ ku
An important observation is that H∗ ku = Z2, and we can identify D∗ with the
augmentation map of H∗MSpinc. We can summarize these maps in more compact
notation by writing H1 = H Σ8∗ BPSp(3) 8+ and H2 = H∗Σ BSU(3)+.
T ∗
µ D
H∗MSpin
c ⊗ (H1 ⊕H2) H∗MSpinc ⊗H MSpinc H MSpinc ∗∗ ∗ H∗ ku
1⊗p
H MSpinc ⊗H MSpinc m∗ ∗ H∗MSpinc
Recall that M = Z2⊗RM , and here p : M → M is x 7→ 1⊗x. In these terms µ is
given by (1⊗x)⊗y →7 1⊗xy, whilem◦(1⊗p) is (1⊗x)⊗y 7→ (1⊗x)⊗(1⊗y) 7→ 1⊗xy.
It follows that the diagram commutes and T ∗ is equal to the composition
m
H c c c c∗MSpin ⊗ (H1 ⊕H2) → H∗MSpin ⊗H∗MSpin −→ H∗MSpin .
Lemma 1.11. To show that T∗ is surjective onto kerD∗, it suffices to show
( )
Hn Σ
8BPSp(3) 4 c+ ∨ Σ BSU(3)+ → QHnMSpin
is surjective for n ≥ 4 with n ̸= 2k ± 1, where QHnMSpinc is generated by indecom-
posable elements in HnMSpinc
23
Proof. First we claim that it suffices to show
( )
H Σ8∗ BPSp(3)+ ∨ Σ4BSU(3)+ → QH∗MSpinc
is surjective. This amounts to the obvious fact that kerD∗ is generated (over Z2) by
elements in H∗MSpinc with at least one indecomposable factor. These generators are
of the form µ(x⊗ y) for some x ∈ H∗MSpinc and y ∈ QH∗MSpin (note that x may
be 1 here). Provided y is in the image of t∗, µ(x⊗y) is in the image of T ∗. The result
now follows from Lemma 1.12.
Lemma 1.12. The projection p : H∗MSpinc → H∗MSpinc induces a homomorphism
Qp : QH∗MSpin
c → QH∗MSpinc such that Qp : QH c cnMSpin → QHnMSpin is an
isomorphism of abelian groups for n ≥ 4 with n ̸= 2k ± 1. Further, QHnMSpinc = 0
for n < 4 or n = 2k ± 1.
Proof. Recall that H c 2∗BSpin = Z2[xi , xj : α(i) < 3, α(j) ≥ 3] and we defined the
subring R = Z [p22 1, p22, p3, p4, . . .]. Here pi is the image of x2i−1 under the Thom
isomorphism, so identifying H∗MSpin with H∗BSpin (as algebras), we can write
R = Z2[x2 21, x3, x7, x15, . . .]. It is now easy to see that H∗BSpinc has generators x2i and
x of H BSpinc except for x2, x2j ∗ 1 3, and x2n−1 for n ≥ 3 (since α(2n − 1) = n). This
shows QHnBSpinc = 0 for n < 4 and n = 2k − 1; to finish the lemma, use the ring
identification H∗BSpinc = Z2[wn : n ≥ 2, n ̸= 2k + 1] to show QHnMSpinc is trivial
for all n = 2k + 1.
Remark. We pause here to make an important note: as an A-module, H∗BSpinc is
the quotient of H∗BSO by the ideal generated by Sq3 ([Stong, §XI]). As an algebra,
one can show with the Adèm relations that H∗BSpinc is the polynomial ring with
generators wi for all i ≥ 2 with i ≠ 2k+1. This identification does not respect the full
A-module structure, however it respects the actions by Q0 and Q1. The analogous
24
fact is true for the Spin case and the actions by Sq1 and Sq2 ([Stolz]). An explicit
example of how this can cause issues is given in remark B.5.
After dualizing to cohomology, theorem 1.7 now follows from the following lemma.
Lemma 1.13. The map
PHnBSpinc → Hn(Σ8BPSp(3) 4+ ∨ Σ BSU(3)+)
is injective for n ≥ 4, n ̸= 2k ± 1.
Proof. The proof of [Stolz, proposition 7.5] given by Stolz easily shows that the
primitive generator yn maps injectively for n ≥ 8, α(n) ≥ 2, since in these degrees
zn is the image of yn under H∗BSpinc → H∗BSpin (the primitive generators zn and
yn are detailed in appendix B). The remaining primitives are y6 and y2k for k ≥ 2.
Since k−1α(6) = 2, y 3 k 26 = s3,3 = w2 + w2w4 + w6. Of course α(2 ) = 1, so y2k = w2 . In
either case, yn maps nontrivially, as witnessed by w2 following theorem C.1.
Lemmas 1.13 and 1.11 together prove theorem 1.8 and thus complete the proof of
theorem 1.7.
25
CHAPTER 2
ALGEBRAIC BACKGROUND
In this section we provide an algebraic overview of the Steenrod algebra and its
dual. We then discuss certain subalgebras and quotient algebras relevant to our
purposes. We will assume all (co)homology has coefficients in Z2 unless otherwise
stated.
2.1 The Steenrod algebra and its dual
We denote the Steenrod algebra as A := H∗HZ2. As an algebra, A is generated
by the Steenrod squares Sqn subject to(the Adem)relations: for m < 2n,∑
Sqm Sqn
n− i− 1
= Sqm+n−i Sqi .
m− 2i
0≤i≤⌊m/2⌋
Now A is a Hopf algebra, and the coproduct ∆: A → A⊗A is cha∑racterized as the
algebra homomorphism satisfying the Cartan formula ∆(Sqk) = i+j=k Sq
i ⊗ Sqj.
We write A∗ for the dual algebra. Milnor established that A∗ is the polynomial ring
(over Z2) with generators ξi of degree 2i − 1 for all i > 0 (we also use the convention
ξ0 = 1). We will abuse notation and reuse ∆ to denote the coproduct in A∑ ∗
:
j
∆(ξk) = ξ
2
i ⊗ ξj.
i+j=k
Similarly, we use µ for the product and χ for the conjugate map in either case. The
latter is characterized by commutativity of the following diagrams.
The Steenrod algebra acts naturally on the Z2-cohomology of any space and sat-
isfies the following properties: for a space X with x, y ∈ H∗X
1. Sqn(x) = x2 if deg(x) = n
2. Sqn(x) = 0 if deg(x) < n
26
A⊗A 1⊗χ A⊗A A 1⊗χ∗ ⊗A∗ A∗ ⊗A∗
A Z2 A A∗ Z2 A∗
A⊗A χ⊗1 A⊗A A χ⊗1∗ ⊗A∗ A∗ ⊗A∗
3. Sq(xy) = Sq(x) Sq(y), where Sq = 1 + Sq1+Sq2+ · · · .
As we mentioned, third property is also called the Cartan formula. Note that
Sq(x) is always a finite sum due to the second property. An instructive application
of the properties of the Sqn is when x is the generator of H∗RP∞. In this case
Sq(x) = 1 + x+ x2, and this completely determines the action of A on H∗RP∞. Via
the splitting principle, one can use this to determine the action on BO.
As an algebra, A is generated by the classes Sqn for n ≥ 0, but this is not a
minimal generating set: for example, Sq3 = Sq1 Sq2. We demonstrate now that A is
n
generated (as an algebra) by only the classes Sq2 for all n. First we need a useful
fact of arithmetic modulo 2.
Lemma 2.1. Given positive integers a, b with base-2 expansions a = a(0 +) 2a1 + · · ·+
2nan and b =( b0 + 2b1 + · · ·+ 2
nbn, the m)odulo(-2 b)in(omi)a(l coe)fficie(nt
a
) factors asb
a0 + 2a1 + 2
2a2 + · · ·+ 2nan a0 a1 a2
= · · · an .
b0 + 2b + 221 b2 + · · ·+(2)nbn b0 b1 b2 bn
Proof. By definition or otherwise, a is the coefficient of xb in (1 + x)a. Now the
b
Frobenius map gives
2 n
(1 + x)a = (1 + x)a0((1 + x)2)a2((1 + x)2 )a2 · · · (1 + x)2 aa 2 a ( n)
n
an
= (1 + x)a0 1 + x2 2 1 + x2 2 · · · 1 + x2 .
27
It follows from the Euclidean division algorithm that the(o)nly way to obtain a xb
term is if eac(h )factor k(1 + x2 )ak contributes kx2 bk . Hence a is nonzero precisely ifb
every factor ai is nonzero.
bi ( )
Notice that a factor ai is 0 if a(nd) only if ai = 0 and bi = 1. Hence a usefulbi
interpretation of lemma 2.1 is that a = 1 if and only if every digit in the base-
b
2 expansion of a is greater than or equal to the corresponding digit in the base-2
expansion of b.
n
Theorem 2.2. As an algebra, A is generated by the classes Sq2 for all n.
Proof. If n is not a power of 2, pick the largest integer k such that 2k < n. Let b = 2k
and a = n− b. Then
∑( )b− c− 1
Sqa Sqb = Sqa+b−c Sqc .
( ) a− 2cc ( )
When c = 0, we have the term b−1 Sqa+b Sq0 = b−1 Sqn. Now b − 1 = 2k − 1 has
a a
the base 2 expansion 1+2+22+ · · ·+2k−1, and, by const(ruct)ion, a = n− b can only
contain summands 2i with i ≤ k − 1. By corollary (2.1), b−1 = 1, and thus
∑( ) a
a b n b− c− 1Sq Sq = Sq + Sqa+b−c Sqc .
a− 2c
c>0
That is, ∑( )k
Sqn = Sqn−2
k
Sq2
k 2 − c− 1
+ Sqn−c Sqc .
n− 2k − 2c
c>0
k
Thus Sqn can be written in terms of Sq2 and Sqi with i < n. To avoid infinite
j
descent, we see that Sqn can be written using only Sq2 for various j.
Given a sequence I = (i1, i2, . . . , ir), we use the shorthand SqI = Sqi1 Sqi2 · · · Sqir .
The sequence I is called admissible if ij ≥ 2i Ij+1 for all j; hence the monomials Sq
28
are precisely those with no nontrivial Adém relations. It follows from the Adém re-
lations that A is generated as a vector space by the monomials SqI for all admissible
sequences I. Serre established the linear independence of these classes, hence they
form a basis known as the Serre-Cartan basis for A.
The coproduct also has a particularly elegant description using the SqI notation:
given any sequence I = (i1, . . .(, in)), we h∑ave
∆ SqI = SqI1 ⊗ SqI2 , (2.1)
I1+I2=I
where I1 + I2 denotes component-wise addition. This formula becomes clear upon
examination: applying ∆ to each factor, the terms in ∆(SqI) correspond to a choice
of one term from each ∆(Sqij). These correspond to a choice of number aj between 0
and ij which indicates the term Sqaj ⊗ Sqbj in ∆(Sqij (where aj + bj = ij). Hence the
terms of SqI correspond to any and all choices of sequences which are component-
wise between (0, 0, . . . , 0) and (i1, i2, . . . , in). Each such sequence of course has a
complimentary sequence such that their sum is I.
Remark 2.3. In the coproduct formula (2.1), it is worth emphasizing that none of I, I1,
or I2 are required to be admissible. Also, we are adding sequences together rather than
concatenating: for example, ∆(Sq3,5,2) will a priori have 72 terms including Sq3,5,2⊗1,
Sq1,3,2⊗ Sq2,2,0, and Sq1,1,1⊗ Sq2,4,1. Of course, many of these terms vanish due to
nontrivial relations. The second and third terms mentioned vanish since Sq1,3,2 = 0
and Sq1,1,1 = 0.
2.2 A closer look at A∗
One must be careful when taking the dual of an infinite dimensional space.
Here, A∗ is the subspace of the HomZ2(A,Z2) generated by functionals with finite-
29
dimensional support. Since A is infinite dimensional, the true linear dual has strictly
larger dimension, but we nonetheless refer to A∗ as simply “dual” to A and vice versa.
This is possible since, by only allowing functionals with finite dimensional support,
we ensure that A∗ is isomorphic to A as a graded vector space.
Define I = (2n−1, 2n−2n , . . . , 4, 2, 1) for n > 0 and I0 = (0) (that is, SqI0 = Sq0 =
1). For all n ≥ 0, one can define ξn as the linear dual to SqIn with respect to the
Serre-Cartan basis for A. That is, ⟨ξn, SqI⟩ is 1 if I = In and is 0 if I is any other
admissible sequence. Of course, this also defines ⟨ξ , SqIn ⟩ for any I after applying the
Adem relations. We will write ξ0 = 1. We noted earlier that cocommutativity of A
ensures that A∗ is commutative. Some relatively simple combinatorics will show that,
for every n, the number of admissible sequences of degree n corresponds bijectively
to the number of partitions of n using the numbers 2k − 1 = deg(Ik). This turns
out to not give false hope: Serre showed that, as an algebra, A∗ is the polynomial
ring Z2[ξ1, ξ2, . . .]. Note that deg(ξ ) = 2nn − 1, and as with the notation SqI , we
write ξJ = ξj11 ξ
j2
2 · · · ξjrr for any a sequence J = (j1, . . . , jr) of nonnegative integers.
Note that using admissible sequences correspond to Serre-Cartan basis elements for
A, but for A∗ it is more convenient to use arbitrary sequences (with finite length and
nonnegative integer entries).
The comultiplication on A∗ is given by
∑n
∗ 2i ⊗ ⊗ 2 ⊗ 4 ⊗ · · · 2n−1µ (ξn) = ξn−i ξi = ξn 1 + ξn−1 ξ1 + ξn−2 ξ2 + + ξ1 ⊗ ξn−1 + 1⊗ ξn.
i=0
Sometimes it is convenient to use the Hopf conjugates ζn := χ(ξn). Then the coprod-
30
uct is
∑n
i
µ∗(ζ ) = ζ ⊗ ζ2 = 1⊗ ζ + ζ ⊗ ζ2 4n i n−i n 1 n−1 + ζ2 ⊗ ζn−2 + · · ·+ ζ ⊗ ζ2
n−1
n−1 1 + ζn ⊗ 1.
i=0
In theory, one can compute the coproduct directly by dualizing from the product
on A. Consider µ∗(ξ2), for example. By definition, ξ2 is dual to Sq2,1. There are only
two vector space generators in degree 3, and we have
〈 〉 〈 〉
ξ2, Sq
2,1 = 1 and ξ 32, Sq = 0.
The basis elements which may be summands of µ∗(ξ2) are
ξ2 ⊗ 1, ξ3 2 2 31 ⊗ 1, ξ1 ⊗ ξ1, ξ1 ⊗ ξ1 , 1⊗ ξ1 , and 1⊗ ξ2,
and we can simply check which of these are summands in by evaluating against basis
elements in the Serre-Cartan basis elements Sq3⊗1, Sq2,1⊗ Sq1, etc. For example,
〈 ∗ 〉 〈 ( )〉 〈 〉µ (ξ ) , Sq3⊗1 = ξ , µ Sq3⊗1 = ξ , Sq32 2 2 = 0.
In this fashion we see that µ∗(ξ2) evaluates to 1 on precisely the generators Sq2,1⊗1,
Sq2⊗ Sq1 and 1 ⊗ Sq2,1. By examining the lower dimensions one finds µ∗(ξ2) =
ξ ⊗ 1 + ξ22 1 ⊗ ξ1 + 1⊗ ξ2.
We defined ξn to be dual to Sqn in the basis consisting of monomials SqI with I
admissible; now we set Sq(I) = Sq(i1, i2, . . . , in) to be dual to ξI = ξi1ξi21 2 · · · ξinn with
respect to the basi∑s consisting of all monomials ξJ for any sequence J . Note that
the degree of ξI is jj ij(2 − 1), and we can obtain an element of this degree in the
original basis of A as follows: create an admissible sequence I ′ such that the excess
i′ − 2i′ ′j j+1 in position j is equal to ij. One might hope that SqI is equal to Sq(I), but
this is not true in general. However, we can order these bases to obtain a bilinear
31
′
pairing which recovers SqI from Sq(I) and vice versa. This ordering on both the sets
of admissible sequences and the sets of sequences is lexicographic from the right. For
example,
(100, 21, 3) < (42, 22, 3) < (16, 8, 4) < (8, 4, 2, 1).
2.3 Pairing between A and A∗
Let I be the set of finite sequences of nonnegative integers and let J ⊂ I be the
set of such sequences which are admissible. We give I and J total orders using the
lexicographic ordering from the right. That is, longer sequences have higher order,
and, if two sequences have the same length, we compare the rightmost entries where
they differ (larger means higher order). For example,
(5) < (3, 1) < (4, 1) < (0, 2) < (0, 0, 1).
Note that this order applies to both I and J . We also have an order-preserving
bijection σ : J → I given by σ(j1, . . . , jn) = (j1 − 2j2, j2 − 2j3, . . . , jn−1 − 2jn, jn).
Lemma 2.4. Let I and J be admissible sequences. If I ≥ J , then〈 〉 
ξσ(I), SqJ = 1, if I = J0, if I > J.
Proof. Assuming I is nontrivial, we can write I = (i1, . . . , in) with in > 0. Then we
can subtract 1 from the last entry of σ(I) to get Ĩ := (i1 − 2i2, i2 − 2i3, . . . , in−1 −
2in, in − 1), and we have
〈 〉 〈 ( ) 〉
ξσ(I), SqJ = 〈∆∗ ξ Ĩ( ⊗ ξ)n〉 , Sq
J
= ξ Ĩ ,∆ SqJ
32
∑ 〈 〉
= ξ Ĩ ⊗ ξ , SqJ1 ⊗ SqJ2n .
J1+J2=J
Since ξn is dual to In := (2n−1, 2n−2, . . . , 4, 2, 1), the only possible nonzero term here
is when J2 = In. Thus
∑ 〈 〉 〈 〉〈 〉
ξ Ĩ ⊗ ξ , SqJ1 J2 Ĩ J−In Inn ⊗ Sq = ξ , Sq ξ〈 〉 n
, Sq
J1+J2=J
= ξ Ĩ , SqJ−In .
At this point, notice that J must have at least length n for J − In to be nonnegative.
In addition, the assumption I ≥ J means the length of J is at most n, so we can
write J = (j1, . . . , jn) (note that I ≥ J also implies in ≥ jn). If I = J , then we can
repeat this process until reaching ⟨ξ , SqIkk ⟩ = 1 for some minimal k. We now assume
I > J . It’s easy to see that J − In = (j − 2n−1, j n−21 2 − 2 , . . . , jn−1 − 2, jn − 1) is
admissible, and since in ≥ jn, we can repeat this process jn−1 more times. If in = jn,
we start from the beginning, replacing Ĩ with (i1 − 2i2, . . . , in−1 − 2in) and J with
(j n−1 n−21−2 jn, j2−2 jn, . . . , jn−2−4jn, jn−1−2jn). In this case, in−1−2in ≥ jn−1−2jn,
′
and we continue as long as possible. Eventually, we reach some value ⟨ I′ξ , SqJ ⟩ where
the length of I ′ is strictly larger than the length of J ′. As we saw, in this case
⟨ I′ξ , SqJ ′⟩ = 0, which completes the proof.
2.4 The Hopf algebra antipode map and Hopf subalgebras
Since A∗ is also a Hopf algebra, it has an antipode map χ : A∗ → A∗ characterized
by the commutativity of the following diagram.
The diagonal maps are the obvious product and coproduct maps, and the unlabeled
horizontal maps are the augmentation (counit) and unit ma∑ps. For n ≥ 0, the middle
row takes ξn to 0. Comparison with the top row shows 2
j
i+j=n ξi χ(ξj) = 0. In
33
A∗ ⊗A
1⊗χ
∗ A∗ ⊗A∗
A∗ Z2 A∗
A χ⊗1∗ ⊗A∗ A∗ ⊗A∗
particular χ(ξn) is determined inductively via
n−1 n−2
χ(ξn) = ξ
2
1 χ(ξ ) + ξ
2 χ(ξ 4 2n−1 2 n−2) + · · ·+ ξn−2χ(ξ2) + ξn−1χ(ξ1) + ξn.
Adopting the semi-standard convention ζn := χ(ξn), we equivalently have∑n−1
n−i
ζ = ξ + ξ2 ζ 2 4 2
n−1
n n i n−i = ξn + ξn−1ζ1 + ξn−2ζ2 + · · ·+ ξ1 ζn−1.
i=1 ∑
For x ∈ A, we write ∆(x) = x′ ⊗ x∑′′i i i . The fact that A∑is connected means
that, if x ∈ A+, then the conjugate satisfies χ(x′)x′′i i i = 0 = i x′iχ(x′′ +i ), where A
denotes the positively graded part of A. In this case, ∆(x) is the sum of 1⊗x+x⊗1
plus terms in A+ ⊗ A+, hence one can inductively determine χ(x) by this prop-
erty. For example, with the Steenrod algebra we have ∆(Sq1) = Sq1⊗1 + 1 ⊗ Sq1,
hence µ(χ(Sq1) ⊗ 1) + µ(χ(1) ⊗ Sq1) = 0 and χ(Sq1) = Sq1. Next ∆(Sq2) =
Sq2⊗1 + Sq1⊗ Sq1+1 ⊗ Sq2 and so χ(Sq2) = χ(Sq1) Sq1+Sq2 = Sq2. For Sq3
the antipode is nontrivial: χ(Sq3) = χ(Sq2) Sq1+χ(Sq1) Sq2+Sq3 = Sq2 Sq1.
We say E ⊆ A is a sub-Hopf algebra of A if E is a Hopf algebra whose structure
maps are restrictions of those for A. In particular, E is a subalgebra of A, the dual
E∗ is a sub-coalgebra ofA∗, and the conjugation maps preserve E and E∗ as subspaces.
The correct notion of the quotient of A by a Hopf subalgebra E is a bit subtle:
since E is unital, the quotient by the left or right ideal generated by E gives the zero
34
ring. Instead, we use the semi-standard notation A/E to denote the Hopf algebra
quotient Z2 ⊗E A. In light of this, the dual notion involves the cotensor product.
That is, we say Z2 ⊗E A is dual to Z2□EA∗.
2.5 A brief review of comodule theory
We now include a superficial review of comodules and cotensor products. Given
an algebra A over a base ring k, a right A-module M has the structure of a map
µM : M ⊗k A → M ; a left A-module has µN : R⊗k N → R. Completely dual to this,
a right A-comodule C has coaction map ∆C : C → C ⊗k A, and a left A-comodule D
similarly has ∆D : D → A⊗k D. Now the tensor product M ⊗A N may be defined as
the cokernel of
⊗ ⊗ −µ−M−⊗−1−−−1−⊗µM A N −→N M ⊗N,
where the unadorned tensor ⊗ denotes ⊗k. Dually, the cotensor product C□RD is
the kernel of
⊗ −∆−C−⊗−1−−−1⊗C D −∆−→D C ⊗ A⊗D.
As is consistent with the literature, we will refer to elements Qi ∈ A defined as
the linear dual to ξi with respect to the monomial basis. The following facts can be
found in [Rog] 15.5.
A ⟨ 1 2 4 2nLemma 2.5. For all n, (n) := Sq , Sq , Sq , . . . , Sq ⟩ and E(n) := ⟨Q0, Q1, . . . , Qn⟩
are sub-Hopf algebras of A. Moreover, E(n) is the exterior algebra on Q0, . . . , Qn.
The quotient A/A(n) = Z2 ⊗[A(n) A is dual to ]
Z n+1 n□ A = Z ξ2 , ξ2 , . . . , ξ4, ξ22 A(n)∗ ∗ 2 1 2 n n+1, ξn+2, ξn+3, ξn+4, · · · .
The quotient A/E(n) = Z2 ⊗E(1) A is dual to
Z □ A = Z [ξ2, ξ22 E(n)∗ ∗ 2 1 2 , . . . , ξ2 2n, ξn+1, ξn+2, ξn+3, ξn+4, . . .].
35
This lemma is overpowered for our purposes, but it shows A/A(1) is dual to
Z [ξ42 1 , ξ22 , ξ3, ξ4, . . .] = Z2[ζ4 21 , ζ2 , ζ3, ζ4, . . .],
and A/E(1) is dual to
Z [ξ22 1 , ξ22 , ξ3, ξ4, . . .] = Z2[ζ21 , ζ22 , ζ3, ζ4, . . .].
The topological significance is easy to state: for ko the connective cover of the
(real) KO-theory spectrum (and ku the connective cover for complex K-theory),
H∗ ko = A/A(1) and hence H∗ ku = Z2[ξ41 , ξ22 , ξ3, ξ4, . . .].
On the other hand,
H∗ ku = A/E(1) and H∗ ku = Z [ξ22 1 , ξ22 , ξ3, ξ4, . . .].
Note that E(0) = A(0) is generated by 1 and Sq1, and we have also described
H∗HZ = A/A(0) and H∗HZ = Z2[ξ21 , ξ2, ξ3, . . .].
For further reference, see [Adams] (proposition 16.6 of section III), [Stong] (page
330), and [ABP] (page 287).
2.6 A(1) modules and H∗BPSp(3)
Recall that A(1) is the subalgebra of A generated by Sq1 and Sq2. It is easy to
verify that A(1) has basis
{1, Sq1, Sq2, Sq3, Sq2,1, Sq3,1, Sq4,1+Sq5, Sq5,1}.
Here we have written elements in terms of SqI with admissible I, but note that we
have several nontrivial relations: for example,
Sq1,2 = Sq3
36
Sq2,2 = Sq3,1
Sq2,1,2 = Sq2,3 = Sq4,1+Sq5
Sq2,2,2 = Sq2,1,2,1 = Sq1,2,1,2 = Sq5,1 .
We can depict A(1)-modules as graphs with a node for each basis element, a
short edge indicating left multiplication by Sq1, and a long edge indicating left mul-
tiplication by Sq2. Diagrammatically, we indicate degree with vertical position. In
addition to the regular A(1)-module, we consider Z2 as a module concentrated in de-
gree zero. Write I for the augmentation ideal of A(1). The inclusion I ↪→ A(1)
and augmentation A(1) → Z2 maps are of course maps of A(1)-modules. The
other useful A(1)-modules are the “joker” J := A(1)/A(1)(Sq3) and the module
K := A(1)/A(1)(Sq1, Sq2 Sq3). These submodules are depicted in figure 2.1.
6
5
4
3
2
1
0
−1 A(1) I Σ2J K Z2
0 1 2 3 4 5 6 7 8 9 10 11 12 13
Figure 2.1. Relevant submodules of A(1).
A key computation in [Stolz] was the the A(1)-module structure of H∗BPSp(3).
37
We revisit Stolz’s computation beginning with theA(1) action onH∗BPSp(3) ([Stolz]).
We will rely on Kono’s determination in [Kono] that
H∗BPSp(3) = Z2[t2, t3, t8, t12],
where deg ti = i. To compute the the actions of Sq1 and Sq2 on generators, Stolz
used representations of the subgroup P(Sp(1)3) ⊂ PSp(3) using the geometric fact
that Sp(1) naturally acts on H as the unit quaternions. He found that Sq1 and Sq2
act as follows:
Sq1(t 12) = t3 Sq (t3) = 0 Sq
1(t8) = 0 Sq
1(t12) = 0
Sq2(t 2 2 2 22) = t2 Sq (t3) = t2t3 Sq (t8) = 0 Sq (t12) = t2t12.
Lemma 2.6. As A(1)-modules, H∗BPSp(3) ∼= Z2[t8, t212]⊗(Z2[t2, t3]⊕ Z2[t2t12, t3t12]).
Proof. The Cartan formula shows
Sq1(xy) = Sq1(x)y+x Sq1(y) and Sq2(xy) = Sq2(x)y+Sq1(x) Sq1(y)+x Sq2(y),
hence to verify that Z 22[t8, t12] is an A(1)-submodule, it suffices to check that genera-
tors Sq1 and Sq2 take generators t8 and t212 to Z2[t 28, t12]. This is immediate by Stolz’s
computation. The same is true for Z2[t2, t3]. Finally for Z2[t2t12, t3t12] we check
Sq1(t t ) = Sq12 12 (t2)t12 + t Sq
1
2 (t12) = t2t12
Sq2(t2t12) = t
2 2
2t12 + t2t12 = 0
Sq1(t3t12) = 0
Sq2(t3t12) = t2t3t12 + t2t3t12 = 0.
Clearly we have an injection Z 22[t8, t12] ⊗ (Z2[t2, t3]⊕ Z2[t2, t3]t12) → Z2[t2, t3, t8, t12].
Since these modules are finite in each degree we can verify that all elements have been
38
accounted for with generating functions. A generating function for Z2[t2, t3, t8, t12] is
1
(1− x2)(1− x3)(1− x8)(1− x12)
. On the other hand, for Z [t , t22 8 12]⊗ (Z2[t2, t3]⊕ Z2[t2, t3]t12) we have the generating
function ( )
1 1 x12
+
(1− x8)(1− x24) (1− x2)(1− x3) (1− x2)(1− x3)
( )
1 1 + x12
=
(1− t8)(1− t12)(1 + t12) (1− x2)(1− x3)
1
=
(1− x2)(1− x3)(1− x8)(1− x12)
Next we aim to write Z2[t8, t212], Z2[t2, t3] and Z2[t2t12, t3t12] in of A(1), Z2, I, J ,
and K. The utility of 2.6 is that Z2[t8, t12] has trivial action by Sq1 and Sq2, hence
is the direct sum of Z2 with a trivial module. This reduces our task to finding the
A(1)-module structure of Z2[t2, t3] and Z2[t2t12, t3t12]. We refer the reader to [Stolz,
proposition 6.5] for the remainder of the proof.
Theorem 2.7 (Stolz lemma 7.6). Suppose f : C → D is a map of left A(1)-modules,
where C is a direct sum of suspensions of Z2, J , and A(1), and D is a direct sum of
suspensions of Z2, I, J , K, and A(1). If f is injective and f induces an injection
H(C;Q0) → H(D;Q0), then f is a split injection.
2.7 E(1) modules and H∗BSU(3)
Let E(1) be the subalgebra of A generated by Q0 = Sq1 and Q1 = Sq2 Sq1+Sq3.
Note that E(1) is a subalgebra of A(1), and the inclusion E(1) → A(1) yields a
39
restriction of scalars functor A(1)-Mod ⇒ E(1)-Mod right adjoint to induction
A(1)⊗E(1) − : E(1)-Mod ⇒ A(1)-Mod. As before we consider Z2 as a submodules
of E(1) concentrated in degree 0. Let L be the augmentation ideal of E(1) and let
C = E(1)/E(1) Sq1. These E(1)-modules are shown in figure 2.2
4
3
2
1
0
−1 E(1) L C Z2
0 1 2 3 4 5 6 7 8
Figure 2.2. Relevant submodules of E(1).
Lemma 2.8. We have isomorphisms of E(1)-modules
A(1) ∼= E(1)⊕ Σ2E(1)
I ∼= L⊕ Σ2E(1)
J ∼= Z −22 ⊕ Σ E(1)
K ∼= Σ−1L.
Lemma 2.9 (Stolz proposition 6.5). As an A(1)-module, H∗BPSp(3) is the direct
sum of a free module with a direct sum of 8i-fold suspensions of Z , Σ−1I, Σ42 J , and
Σ4K.
Corollary 2.10. As an E(1)-module, H∗BPSp(3) is a direct sum of 4i-fold suspen-
sions of Z2 and Σ−1L.
40
Proof. Thus lemma 2.9 shows that H∗BPSp(3) is, as an E(1)-module, the direct sum
of a free module with a direct sum of 8i-fold suspensions of Z , Σ−12 L, Σ4Z2, and
Σ3L.
As an algebra, H∗BSU(3) is the polynomial ring Z2[y4, y6], where deg(yi) = i. In
theorem A.2 we compute the actions by Sq1 and Sq2, and we summarize the result
here: Sq1(y 2 16) = Sq (y6) = Sq (y4) = 0 and Sq2(y4) = y6. Hence Z2[y24, y6] is an E(1)-
submodule of H∗BSU(3) with trivial action. We can now make a better description
of H∗BSU(3).
Theorem 2.11. As an E(1)-module, H∗BSU(3) ∼= Z2[y24]⊕C ⊗ 2Z2 Z2[y4, y6]y4. Here
Z2[y24] and Z2[y24, y6]y4 are trivial E(1)-modules which may be considered as submod-
ules of Z ∼ ∗2[y4, y6] = H BSU(3).
Proof. Knowing the actions of Sq1 and Sq2 on generators y4 and y6, it is easy to verify
Z2[y24] is a trivial submodule of Z2[y4, y6]. As vector spaces,
Z [y , y ] ∼2 4 6 = Z 22[y4]⊕ Z [y22 4, y6]y4 ⊗ C.
To see this, observe that a generating function in t for the right hand side is
1 t4(1 + t2) 1− t6 + t4(1 + t2)
+ =
1− t8 (1− t8)(1− t6) (1− t4)(1 + t4)(1− t6)
1
= .
(1− t4)(1− t6)
To make this explicit, momentarily write {1, x} for a Z2-basis of C and define a map
φ : Z 2 22[y4]⊕ Z2[y4, y6]y4 ⊗ C → Z2[y4, y6]
by extending linearly from φ(θ , θ ⊗ 1) = θ + θ and φ(θ , θ ⊗ x) = θ + θ y−11 2 1 2 1 2 1 2 4 y6.
Since the domain and codomain of φ are finite dimensional in each grading, φ is a
linear isomorphism provided φ is surjective. Elements of the form y2m4 yn6 are obtained
41
as φ(0, y2m n−1 2m4 y6 y4 ⊗ x) if n ≥ 1 and φ(y4 , 0) if n = 0. For odd powers of y4, observe
y2m+1yn = φ(0, y2m+1 n4 6 4 y6 ⊗ 1). Hence φ is an isomorphism of graded vector spaces
2.8 ko and ku
Let ko be the connective cover of KO, meaning there is a map ko → KO inducing
isomorphisms πn ko → πn KO for n ≥ 0 and where πn ko = 0 for n < 0. Similarly one
may define ku as the connective cover of the complex K-theory spectrum K. Adams
notes that one may take the zeroth space ku to be Z×BU, hence ku0(X) = K0(X) for
any CW complex space X; in general the higher cohomology groups may differ. The
similar statements apply for ku. The α and αc genus correspond to spectrum maps
MSpin → ko and MSpinc → ku, and for this reason we care about the A-module
structures of ko and ku.
As an A-module, we can identify H∗ ko = A/A(1) and H∗ ku = A/E(1). Thus
H∗ ko andH∗ ku are as given in lemma 2.5. We prove the result for ku here, following a
proof by Bruner ([Bru]) which in turn was adapted from Adams: for further reference,
see proposition 16.6 of section III of [Adams].
Lemma 2.12. As an A-module, H∗ ku ∼= A/E(1).
Proof. We begin with the fibration induced by ku → HZ. The total space is Σ2 ku,
giving a cofibration which extends to the right as in Σ3 →−i →−j kku HZ →− Σ3 ku. We
compare the two fiber sequences below.
Σ2 ku ∗
Σ2 ku i ku Σ3 ku
j
k
HZ
42
Write ω ∈ HnΣnn ku for the fundamental cohomology class. The transgression in
the upper right fibration is τ(ω ) = ω . It follows that k∗2 3 ω3 = τω2. We compute
τ(ω2) by looking at the third space in each spectrum. Recall that
ku = (BU× Z,U,BU, SU,BSU, SU⟨5⟩, SU⟨6⟩, . . .)
and
HZ = (Z, S1,CP∞, K(Z, 3), K(Z, 4), . . .).
Restricting to the third spaces gives SU⟨5⟩ → SU → K∧(Z, 3). Write ι ∈ H33 K(Z, 3)
for the fundamental class. We also know that H∗SU = (e3, e5, e7, . . .). Since SU⟨5⟩
is 4-connected, the first nonzero cohomology group H5SU⟨5⟩ = H5SU is generated
by e5. We also know H∗K(Z, 3) (for example using CP∞ → ∗ → K(Z, 3)): the first
several generators are 1, ι3, Sq2 ι3, and Sq3 ι3 = ι23. Since H5SU is one-dimensional
(and generated by e5), we necessarily have the transgression τ(e5) = Sq3 ι3. Now we
turn to the cofibration ku → HZ → Σ3 ku. Write u ∈ H0HZ for the fundamental
class. We have an isomorphism α : A/A Sq1 → H∗HZ given by α(θ) = θu and a
diagram In addition, we have the exact sequence
A/A Sq1
α
j∗ ∗
0 H∗Σ3 ku H∗(HZ) H∗ ku i 0
0 → A f/(Sq1, Sq3) →− A/(Sq1) →−g A/(Sq1, Sq3) → 0
where f(x) = x Sq3 and g is the quotient: note that f is well defined since Sq3 Sq3 = 0
and Sq1 Sq3 = 0, and obviously im f ⊆ ker g. On the other hand ker g is gener-
ated by elements of the form x Sq1+y Sq3, where x, y ∈ A. Since f(y) = y Sq3 =
x Sq1+y Sq3 ∈ A/(Sq1), the sequence is exact. Our first computation shows that k∗
hits Sq3, so j∗α induces the map β. Together we get the following diagram.
43
A f g f( /(Sq1,Sq3)) 1n−3 (A/(Sq )) (A/(Sq1n ,Sq3)) δn (A/(Sq1, Sq3))n−2 (A/(Sq1))n+1
β α β β α
∗
Hn−
∗ ∗
3 ku k
j
Hn(HZ) Hn ku δ Hn−2 ku k Hn+1(HZ)
We know that α is an isomorphism, and the Hurewicz theorem shows that the
map β : A/(Sq1, Sq3) → Hnn ku is an isomorphism for n ≤ 2. If β is an isomorphism
for n ≤ r with r > 2, it follows from the five-lemma that the middle vertical map in
the diagram is an isomorphism. By induction the result follows.
2.9 Extended modules and indecomposable quotients
As a general fact, if A is an algebra over a ring B and N is a left B-module,
A⊗B N is a left A-module in the obvious way. Further, if N is free as an B-module,
then A ⊗B N is free as an A-module (the same is true for projective modules). In
this context, we call A⊗B N an extended B-module. In practice, B may be easier to
work with than A, so it is a desirable property for an A-module to be an extended
B-module.
In addition to free modules, extended modules arise in the more interesting case
that A is a Hopf algebra and B is a sub-Hopf algebra of A. Two other notable ex-
tended modules are H∗ ko = A⊗ Z and H∗A(1) 2 ku = A⊗E(1) Z2.
Let A and B be a sub-Hopf algebras of A with B ⊆ A, and let N be a left B-
module. The tensor product A⊗B N is naturally a left A-module in the obvious way.
If B = Z2, then A⊗B N is a free A-module: generators of N over Z2 each generate a
copy of A in A⊗B N . This is true more generally: if N is a free as a left B-module,
then A⊗B N is a free left A-module. Lemma 2.13 is essentially an observation which
appears in [Stolz].
44
Lemma 2.13. The map D : MSpin → ko induces a split surjection on homology
which allows us to consider H∗ ko as a subalgebra of H∗MSpin.
Lemma 2.14. ([Stolz, corollary 5.5]) Let Y be an MSpin-module spectrum whose ho-
mology is bounded below and locally finite. Identify H∗ ko as a subalgebra of H∗MSpin
as in lemma 2.13. Then there is a functorial isomorphism
H∗Y → A∗□A(1)∗H∗Y ,
where H∗Y := Z2 ⊗H∗ ko H∗Y . Given a map f : X → Y of MSpin-module spectra,
we can identify f∗ : H∗X → H∗Y with 1 ⊗ f ∗, where f ∗ is a map H∗X → H∗Y . In
addition, if f ∗ is a split surjection of A(1)∗-comodules, then f∗ is a split surjection of
A∗-comodules.
Lemma 2.15. Let Y be an MSpinc-module spectrum whose homology is bounded
below and locally finite, and let R ⊂ H∗MSpinc be the subalgebra which D∗ maps
isomorphically to H∗ ku. Let H∗Y := Z2 ⊗R HY be the R-indecomposable quotient of
H∗Y . Then there is a functorial isomorphism of A∗-comodules
H∗Y → A∗□E(1)∗H∗Y .
Given a map f : X → Y of MSpinc-module spectra, we can identify f∗ : H∗X → H∗Y
with 1⊗ f ∗, where f ∗ is a map H∗X → H∗Y . In addition, if f ∗ is a split surjection
of E(1)∗-comodules, then f∗ is a split surjection of A∗-comodules.
Lemmas 2.14 and 2.15 are both particular cases of [Stolz, proposition 5.4], so we
defer to his proof.
45
CHAPTER 3
TRANSFER MAPS
Unless specified otherwise, we will assume that all spaces have a cell structure
with finitely many cells in each dimension. For two spaces X, Y , we write [X, Y ]
for homotopy classes of maps X → Y . Given based spaces (X, x0) and (Y, y0), we
similarly write [(X, x0), (Y, y0)] for (basepoint preserving) homotopy classes of maps
(X, x0) → (Y, y0). The notation X+ means the disjoint union of X with a point.
3.1 Virtual vector bundles
Define a virtual vector bundle over X to be a pair (X, f), where f : X → BO×Z
is an unbased map. For convenience, we sometimes omit the word “vector” and say
(X, f) is a “virtual bundle.” Homotopy classes of maps X → BO × Z form the
(unreduced) real K-theory of X; i.e., KO(X) := [X,BO×Z]. (Note that KO(X) can
equivalently be defined as based homotopy classes of maps X+ → BO× Z.) Given a
virtual bundle ξ = (X, f), we call f the classifying map of ξ. Projection BO×Z → Z
defines the rank of ξ. When X is connected, the rank may be considered an integer
rk(ξ) ∈ Z. Two virtual bundles (X, f) and (X, g) are equivalent if they correspond
to the same class of KO(X) (i.e., if f and g are homotopic).
By contrast, a genuine vector bundle η over a connected space X is classified by
a map X → BO(n), where n is the rank of the bundle. Now η yields a virtual bundle
of rank n via the composition
X → BO(n) → BO× {n} ↪→ BO× Z.
Generally we abuse notation and write η for both the vector bundle and virtual
bundle. However, it is important to note that a virtual bundle does not always arise
from a genuine vector bundle in this way. For example, virtual bundles can have
46
negative rank, and, if X is not compact, a map X → BO×Z may not factor through
BO(n)× Z for any n.
While there are virtual bundles which do not arise from vector bundles, there is
also some loss of information in considering vector bundles as virtual bundles. For
example, a stably trivial vector bundle yields a trivial virtual bundle of the same
rank: suppose ξ ⊕ k ∼= n ⊕ k, where n is the rank of ξ and k is the trivial vector
bundle of rank k. Then X →−f BO(n) ↪→ BO(n+ k) is nullhomotopic, so the resulting
composition X → BO is also nullhomotopic. This means that ξ, considered as a
virtual bundle, is equivalent to the trivial virtual bundle of rank n.
Remark 3.1. It is immediate from the definitions that trivial virtual bundles are
equivalent if and only if they have the same rank. It follows that vector bundles ξ
and η over a connected base space X yield equivalent virtual bundles if and only if
rk(ξ) = rk(η) and ξ ⊕ k ∼= η ⊕ k for some k.
Suppose we have a genuine vector bundle ξ of rank n over a CW complex X.
In particular, X is paracompact, so we can embed ξ into a trivial vector bundle
k for some k. The complementary bundle ξ⊥ (which can be defined topologically
as the quotient bundle of k by ξ) can be used to form an additive inverse of ξ as
classes in KO(X). That is, KO(X) is an additive group, and, since ξ + ξ⊥ = k,
we have −ξ = ξ⊥ − k. Hence −ξ can be considered as the formal difference ξ⊥ − k
(which is a virtual bundle of rank −n). The (virtual) bundle −ξ is called the stable
complement of ξ. In particular, when X is a manifold, we define the stable normal
bundle of X as the stable complement to the tangent bundle (considered as a virtual
bundle).
Now suppose ξ is stably trivial. Without loss of generality, ξ ⊕ 1 ∼= n+ 1. Then
X →−f BO(n) ↪→ BO(n+ 1) is nullhomotopic, so f̃ : X+ → BO × Z is homotopic to
47
the constant map with image {b0} × {n} and ξ̃ ≡ ñ. This shows that stably trivial
vector bundles become trivial virtual bundles of the same rank. Similar logic shows
that stably equivalent bundles of equal rank represent equivalent virtual bundles (of
the same rank). It is important to not overlook the requirement that the genuine
bundles had the same rank to begin with. For example, trivial virtual bundles of
distinct ranks are not equivalent.
3.2 Thom spectra and Dold’s theorem
Given a CW spectrum X, we can suspend or desuspend X to obtain ΣrX for any
r ∈ Z, where (ΣrX)i = (X)r+i. A map of spectra X → Y of degree d is defined
as a map ΣdX → Y of degree 0 (a map S0 → S0 of degree d, for example, is a
stable map Sd → S0). To avoid mentions of degree, we will instead use suspensions
and desuspensions as appropriate to make all maps have degree 0. For this reason,
assume maps of spectra have degree 0 by default. We will now describe some relevant
spectra.
Example 3.2 (Spectrum of a CW space). Any CW complex X can be considered a
(CW) spectrum whose i-th space (X)i is the basepoint {x0} for i < 0 and (X)i = ΣiX
for i ≥ 0. We use X to denote the spectrum as well as the space.
Example 3.3 (MO and similar Thom spectra). Write γn for the universal n-plane
bundle over BO(n). The inclusion BO(n) → BO(n+1) is induced by the composition
O(n) ↪→ O(n)×O(1) → O(n+ 1), so γ ∼n+1|BO(n) = γn ⊕ 1. The classical construction
of Thom spaces yields spaces MO(n) and maps ΣMO(n) → MO(n+ 1) which give a
spectrum MO. Similar constructions give MSO, MU, MSpin, MSpinc, etc.
48
Example 3.4 (Thom spectrum of a genuine vector bundle). Suppose we have a
(genuine) vector bundle ξ over a CW complex X of rank k. The classifying map
cξ : X → BO(k) induces Xξ → MO(k) (here Xξ denotes the classical Thom space
rather than a spectrum; existence of cξ is detailed in [Switz] 11.33). The pullback of
γk+1 by the composition X → BO(k) → BO(k + 1) yields a map ΣXξ ∼= Xξ+1 →
MO(k + 1), and inductively we get a spectrum map Σ−kXξ → MO. (Equivalently,
we have a map Xξ → MO of degree −k.)
Example 3.5 (Thom spectrum of a virtual bundle represented by a genuine bundle).
Suppose now that ξ is a virtual bundle of (virtual) rank k over a CW complex X such
that the classifying map X → BO factors through BO(n) for some n. This is true,
for example, when X is finite dimensional ([Switz] 6.35). Let α be the induced vector
bundle of rank n. Our first construction yields a classifying map Σ−nXα → MO of
degree 0. Define Xξ := Σk−nXα so as to have the classifying map Σ−kXξ → MO
(just as we did when ξ was a genuine bundle of rank k).
We should ensure that, in an appropriate sense, the spectrum Xξ and classifying
map Σ−kXξ → MO are well-defined regardless of the choice of representative α.
Suppose a (genuine) m-bundle β also represents η. The maps cα, cβ : X → BO
are then homotopic via a cellular homotopy H: to see this, apply relative cellular
approximation to (X× I,X×{0, 1}). This means H factors through BO(ℓ) for some
ℓ; in particular, α ⊕ s is isomorphic (as a vector bundle) to β ⊕ t for some s, t ∈ N.
Hence
Σk−nXα ≃ Σk−n−sXα⊕s ≃ Σk−n−sXβ⊕t ≃ Σk−n−s+tXβ.
Since α⊕s ∼= β⊕t, we have n+s = m+t, so Σk−n−s+tXβ = Σk−mXβ. All equivalences
shown are natural, so we have a natural equivalence Σk−nXα ≃ Σk−mXβ which, after
desuspension, also carries the classifying map Σ−nXα → MO to the classifying map
49
Σ−mXβ → MO.
Starting with a genuine bundle ξ, we can consider ξ to be virtual, in which case
Example 3.4 and Example 3.5 give potentially distinct definitions of Xξ. However, it
is trivial to check that both constructions agree.
Example 3.6 (Thom spectrum of a negative bundle). Let ξ be a genuine vector
bundle of rank k over X. As a virtual bundle, ξ has additive inverse −ξ of rank
−k. Explicitly, since X is paracompact, we can include ξ into a trivial vector bundle
of rank n. The orthogonal complement ξ⊥ is then a genuine vector bundle of rank
n− k. We simply let −ξ be the virtual bundle represented by ξ⊥ and with rank −k
(recall that the rank is just given by BO×Z → Z). Alternatively, and perhaps more
naturally, notice that ξ ⊕ ξ⊥ = n, so −ξ = ξ⊥ − n. The formal difference ξ⊥ − n
then has rank n− k − n = −k. Note that both definitions of −ξ are equivalent: the
presence of a trivial vector bundle does not affect the resulting map X → BO, only
the component X → Z.
For all our purposes, we have seen how a virtual bundle η of rank r defines an
associated Thom spectrum Xη, even when r < 0. In this context, we have the fol-
lowing generalization of the classical Thom isomorphism established by Dold (details
can be found in [Board, §4.4]), as we explain subsequently.
3.3 The general bundle transfer map
Virtual bundles and generalized Thom isomorphisms
As usual, let BO denote the CW complex filtered by the classifying spaces BO(n).
For a connected CW complex X, a virtual bundle ξ over X of rank r ∈ Z is an
50
element of KO(X) := [X+,BO× Z] whose projection onto Z is identically r. We can
now define orientations of a virtual bundle with respect to any oriented spectrum A.
For any virtual bundle ξ over X, we have an associated Thom spectrum Xξ. We
will write n for the trivial (virtual) bundle of rank n overX. For n ≥ 0, Xn = Sn∧X+.
In particular X0 = X+ is the disjoint union of X with a point.
We note a potential confusion in terminology here: a genuine vector bundle ξ
of rank k over X can be considered a virtual bundle of the same rank, but the
classifying map Xξ → MO has degree −k (or codegree k). In this case, ξ has a
genuine classifying map X → BO(k), and we use the same term for the induced map
of spectra Xξ → MO. Classes of maps toMO are more general, since a virtual bundle
cannot necessarily be classified by a map X → BO(n) for some n.
Let A be a spectrum with unit i : S0 → A, and let ξ be a virtual bundle of rank
n over a connected CW complex X. A spectrum map u : Xξ → A of degree −n (or
codegree n) is a fundamental class of ξ (with respect to A) if u restricts to i on
each fiber of ξ. More precisely, for each x ∈ X the fiber of ξ in Xξ can be identified
with Sn. This gives a degree n map of spectra S0 → Xξ (whose 0-th component is a
map Sn → Xξ), and composing with u : Xξ → A gives a map S0 → A of degree 0. If
ξ has a fundamental class with respect to A, we also say ξ is A-oriented. Now we
can state the following generalization (by Dold) of the classical Thom isomorphism
(details can be found in [Board, §4.4]).
Theorem 3.7 (Dold). Suppose a virtual bundle ξ of rank k over a CW complex X
is oriented with respect to a spectrum A with unit i : S0 → A. In addition, let C be a
spectrum with a left A-action A ∧ C → C. Then for any virtual bundle η, there are
“Thom isomorphisms”
Φξ : C̃∗(Xη) → C̃∗+k(Xη+ξ) and Φ η+ξ ηξ : C̃∗(X ) → C̃∗−k(X ).
51
In particular, when η = 0, these isomorphisms become
Φξ : C∗(X) → C̃∗+k(Xξ) and Φξ : C̃∗(Xξ) → C∗−k(X).
When X is a compact smooth manifold with tangent bundle τ , we say that X
is A-oriented if −τ is A-oriented. This definition is (naturally) equivalent to the
more classical version using local homology groups ([Board] 4.7). This also allows
for a notion of A-orientation for certain bundles whose fibers are smooth manifolds.
Suppose π : E → B is a bundle whose fiber F is a closed k-manifold and whose
structure group G is a compact Lie group acting smoothly on F . We say that E
has an A-structure if the virtual bundle −τ is A-oriented, where τ is the bundle of
tangent vectors along the fibers of π and −τ is the stable complement. Thus, an
A-structure on E amounts to appropriately compatible A-orientations of each fiber
π−1(b) (for all b ∈ B).
The Thom map associated to a bundle
Suppose for the remainder of this section that we have a fiber bundle π : E → B
whose fiber F is a closed k-manifold and whose structure group G is a compact Lie
group acting smoothly on F . Again, let τ be the bundle of tangent vectors along the
fibers of π and let −τ denote the stable complement. I.e., τ is a vector bundle over
E whose fiber over x ∈ E is the tangent space of π−1(π(x)) ∼= F at x.
In this setting, we have a transfer T (π) : ΣkB −τ+ → E . We summarize the con-
struction here and refer to [Board] 6.20 for more details. First consider the case when
B is a compact CW complex (then E is compact: it has an induced CW structure
from F and B with finitely many cells). For some d, there is a representation space
Rd along with a G-equivariant smooth embedding F ↪→ Rd ([Board] 6.19). Write
EG for the principal G-bundle underlying π : E → B, and consider the associated
52
Rd-bundle η = E dG ×G R . By compactness, η embeds (via a bundle map) into a
trivial vector bundle B ×Rn+k for some n. Let U be a tubular neighborhood of F in
Rd and consider the bundle EG ×G U . We now have the bundle maps shown.
F U Rd Rn+k
E EG ×G U EG ×G Rd B × Rn+k
Wemay consider EG×GU to be a fiberwise tubular neighborhood of E in B×Rn+k.
In addition, we can identify EG×G U with the unit disk bundle of the normal bundle
ν of E ↪→ B×Rn+k. Collapse the complement of EG×GU to a point and thus obtain
a map Σn+kB → Eν+ on Thom spaces. Note that ν here is a genuine bundle of rank
n over E.
For our purposes, we must modify this construction to apply when B is only
filtered by compact CW complexes B0 ⊂ B1 ⊂ B2 ⊂ · · · . Then each Bi has an
associated bundle νi over E := π−1i (Bi), say with rank ni, and we can ensure these
be compatible in the sense that the νi form a virtual bundle ν (of rank 0) over E.
Then the maps Σni+kB → Eνi form a spectrum-level map ΣkB → E−τ+ + , where −τ
is considered a virtual bundle over E (of rank 0). For more details see [Führ].
Constructing the umkehr map from the Thom map
Definition 3.8 ([Board] V.6.2). Let A be a ring spectrum and let C be a spectrum
with a left A-action A∧C → C. The transfer maps associated to a map f : X → Y
of spaces refer to functorial homomorphisms
f : C∗X → C∗−r! Y and f ! : C∗Y → C∗+rX
which are multiplicative in the sense that f! and f ! are maps of C∗-modules and, for
α ∈ C∗X, β ∈ C∗Y , and y ∈ C∗Y , we have
53
(a) f!(α ⌣ f ∗β) = f!(α) ⌣ β;
(b) f (f ∗(α) ⌣ β) = (−1)r|α|! α ⌣ f!β;
(c) f !(y ⌢ α) = f !(y) ⌢ f ∗(α);
(d) f∗(f !(y) ⌢ α) = (−1)r|y|y ⌢ f!(α);
(e) ⟨f !x, α⟩ = (−1)r|x|⟨x, f!α⟩.
Theorem 3.9 ([Board] V.6.21 and V.6.2). Let π : E → B be a bundle with fiber F
and structure group G satisfying the following:
• E and B are CW complexes;
• F is a compact smooth manifold of dimension k;
• G is a compact Lie group which acts smoothly on F ;
• there is a ring spectrum A and a spectrum C with a left A-action such that −τ
is A-oriented, where τ is the bundle of tangents along the fibers of π.
Then we have transfer maps
π! : C
∗(E) → C∗−k(B) and π! : C∗(B) → C∗+k(E)
which are multiplicative in the sense of definition 3.8 and are the respective composi-
tions
∗ Φ−τ
∗
C (E) −−→ C̃∗ −τ −T−(π(E ) −→) C̃∗(ΣkB+)
and
T (π)∗ Φ−τ
C̃ k −τ∗+k(Σ B+) −−−→ C̃∗+k(E ) −−→ C∗+k(E).
54
The HP2-bundle transfer
Let us recall an earlier observation: PSp(3) acts transitively on S11 ⊂ H3 and
this descends to a transitive action on HP2. The fiber over a point is PSp(2, 1) :=
P(Sp(2)×Sp(1)), so we have a bundle PSp(2, 1) → PSp(3) → HP2. In turn this yields
a bundle π : BPSp(2, 1) → BPSp(3) with fiber HP2 and structure group PSp(3) acting
by isometries on HP2. Let τ be the bundle of tangent vectors along the fibers of π and
write −τ for the stable complement. We now compute some basic (co)homological
properties of this bundle.
Theorem 3.10. For the bundle HP2 → πBPSp(2, 1) →− BPSp(3) with τ the associated
bundle of tangent vectors along the fiber,
1. H∗BPSp(2, 1) ∼= Z[u2, u3, u4, u8], where deg(ui) = i;
2. H∗BPSp(3) ∼= Z2[t2, t3, t8, t12], where deg(tj) = j;
3. π∗(t2) = u2 and π∗(t3) = u3, while π∗(t8) = u24 + u8 and π∗(t12) = u4u8;
4. Sq1(t2) = t3 and Sq1(t 13) = Sq (t4) = Sq1(t12) = 0;
5. Sq2(t ) = t2, Sq22 2 (t
2
3) = t2t3, Sq (t8) = 0, and Sq2(t12) = t2t12;
6. w(τ) = 1 + (u22 + u4) + (u2u4 + u23) + u3u4 + u8;
7. π! : H∗BPSp(2, 1) → H∗−8BPSp(3) is given modulo t12 by a b c/2−1)t3t2t8 , if c > 0 is even and d = 0,
π (uaubucud) = tatbtd−1! 3 2 4 8  3 2 8 , if c = 0 and d > 0,0, if otherwise.
55
We delay the proof of these facts and show how they apply. Our HP2-bundle has a
fiberwise Spin-structure, i.e., −τ is oriented with respect to MSpin. Applying Board-
man’s construction ([Board]) to the fiber sequence HP2 → πBPSp(2, 1) →− BPSp(3)
yields a bundle transfer map T : Σ8BPSp(3)+ → M(−τ). Meanwhile, theorem 3.7
gives identifications
H̃∗M(−τ) ∼= H∗BPSp(2, 1) and H̃∗M(−τ) ∼= H∗BPSp(2, 1).
In terms of spectra, the transfer of theorem 3.9 with C = MSpin is then
MSpin∧Σ8 −1−∧→TBPSp(3)+ MSpin∧M(−τ) → MSpin∧BPSp(2, 1)+,
where the righthand map induces the Thom isomorphism. The Thom isomorphism
can be described more explicitly as the map induced on homotopy groups of the
composition indicated below. We use G = PSp(3) and H = PSp(2, 1); dashed arrows
indicate a map induced on homotopy groups.
Ψ
T
1∧t
µ
MSpin∧Σ8BG 1∧T+ MSpin∧M(−τ) 1∧M MSpin∧MSpin MSpin
π!
Φ
proj
1∧∆
MSpin∧M(−τ) ∧BH+ ∧M∧ MSpin∧MSpin∧BH MSpin∧BH1 1 + µ∧1 +
Here M is the map of Thom spectra induced by the classifying map BH → BSpin
of −τ , which motivates theorem 3.10 (in particular, it demonstrates why computing
w(−τ) is relevant).
As mentioned earlier, we will use the same names for several of these maps
with MSpin replaced with MSpinc. For example, T : Σ8BPSp(3) → MSpinc and
T c : Σ4BSU(3) → MSpinc together are used to prove theorem 1.7.
56
3.4 The CP2-bundle transfer
Analogously to the quaternionic case, the unitary group U(3) acts transitively on
S5 ⊂ C3 yielding a transitive action on CP2. As before, the stabilizer of [0 : 0 : 1] is
SU(2, 1) := S(U(2)×U(1)) and we hence obtain a bundle BSU(2, 1) → BSU(3) with
fiber CP2. This bundle admits a Spinc structure, giving a map BSU(3) → BSpinc.
Now a class in cΩSpinn BSU(3) consists of an n-manifold P with a stable normal Spin
c
structure along with a map f : P → BSU(3). Let P̂ be the pullback of the bundle
BSU(2, 1) → BSU(3); hence P̂ is an (n+ 4)-dimensional Spinc manifold fibered over
P . On bordism classes, this correspondence [P ] → [P̂ ] geometrically describes the
c
transfer map ΩSpincn BSU(3) → Ω
Spin
n+4 .
We have the following maps of spectra.
MSpinc ∧ Σ4 µBG 1∧t+ MSpinc ∧MSpinc MSpinc D ku
T
These induce maps on homology.
c ⊗ 4 −1−⊗→tH MSpin H Σ BG ∗ µ D∗ ∗ + H∗MSpinc ⊗H∗MSpinc →− H∗MSpinc −→∗ ku (3.1)
Recall A∗ = Z2[ξ1, ξ2, ξ3, . . .], where deg(ξi) = 2i − 1 with coproduct
∑n
2i 2 2n−1ψ(ξn) = ξn−i ⊗ ξi = ξn ⊗ 1 + ξn−1 ⊗ ξ1 + · · ·+ ξ1 ⊗ ξn−1 + 1⊗ ξn.
i=0
Alternatively, one can use the Hopf algebra conjugates ζi of ξi, characterized by
∑n
⊗ 2iψ(ζn) = ζ ζ 2i n−i = 1⊗ ζn + ζ1 ⊗ ζn−1 + · · ·
n−1
+ ζ 2n−1 ⊗ ζ1 + ζn ⊗ 1.
i=0
The dual of Sqn = Sq(n) is ξn1 , while Qn = Sq(0, . . . , 0, 1) (with n zeros followed by
a 1) is dual to ξn+1.
57
Note thatH∗HZ2 is also the dualA∗ of the Steenrod algebraA. For the subalgebra
E(1) generated by Q0 and Q ∗1, we have H ku = A/E(1) = Z2 ⊗E(1) A. Dually,
H∗ ku = Z2□E(1)∗A∗.
Theorem 2.15 stated that, if Y is an MSpinc-module spectrum for which H∗Y is
bounded below and of finite type, there is a functorial isomorphism of A∗-comodules
H∗Y → A∗□E(1)∗H∗Y .
Applying theorem 2.15 to (A.1) gives
1⊗t µ D
H∗MSpin
c ⊗H 4 ∗∗Σ BG+ −−→ H∗MSpinc ⊗H∗MSpinc →− H∗MSpinc −→∗ H∗ ku.
However, H∗ ku = Z2, and one can identifyD∗ with the augmentation homomorphism
of H∗MSpinc. It follows that kerD∗ is generated over Z2 by elements which have at
least one nontrivial indecomposable factor. These elements can be written µ(x ⊗ y)
for some x ∈ H ∗MSpinc and y ∈ QH∗MSpinc. (Here, x may be 1, and, for an
algebra A with augmentation ideal I(A), QA denotes the indecomposable quotient
I(A)/µ(I(A) ⊗ I(A)).) Thus to show that T ∗ surjects onto kerD∗, it suffices to
show that the composition H Σ4∗ BG+ → H∗MSpinc → H∗MSpinc → QH∗MSpinc is
surjective.
We have QHnMSpinc ∼= Z2 for n ≥ 2, n ̸= 2k + 1 and QH cnMSpin = 0 otherwise.
Further, the projection QH MSpinc → QH MSpincn n is an isomorphism for n ≥ 4,
n ̸= 2k ± 1, and QHnMSpinc = 0 for [n < 4 or n = 2
k ± 1. We]have
H c β(n) k∗MSpin = Z2 xn : n ≥ 4, n ̸= 2 ± 1 ,
where β(n) = 2 if α(n) < 3 and β(n) = 1 for α(n) ≥ 3 (here α(n) is the number of
nonzero terms in the base-2 expansion of n). Ordering the generators by their lower
indices, [ ]
H MSpinc = Z x2, x2, x2, x2∗ 2 1 2 3 4, x25, x26, x7, x28, x29, x210, x11, . . . .
58
Since R = Z [x2, x22 1 3, x7, x15, . . .], we have
[ ]
H∗MSpin
c = Z ⊗ H c 2 2 2 2 2 2 22 R ∗MSpin = Z2 x2, x4, x5, x6, x8, x9, x10, x11, . . . .
We must show that H 4nΣ BG+ → QHnMSpinc is surjective for n ≥ 4, n ≠ 2k ± 1.
Dually, we must show PHnMSpinc → Hn−4BG+ is injective for n ≥ 4, n ̸= 2k±1. Re-
T (π) M(c)
call that the map BG+ → MSpinc can be decomposed into Σ4BG+ −−→ M(−τ) −−−→
MSpinc, where π : BH → BG is the bundle map, T (π) is the Thom collapse map,
and c : BH → BSpinc is classifies the complement −τ of τ , the bundle along the
fibers of π. The h-space inverse BSpinc → BSpinc provides a homotopy equivalence
MSpinc → MSpinc allowing us to replace M(τ) with M(τ). On cohomology, we have
the maps
n c −c−(τ−→)
∗
n −T−(π)
∗
H MSpin H BH −→ Hn−4BG.
• H∗BH = Z2[x2, x4],
• H∗BG = Z2[y4, y6],
• w(τ) = 1 + x2 + x4,
• π∗y 2 ∗4 = x2 + x4 and π y6 = x2x4,
• we can write H∗BH as a free H∗BG-module with basis 1, x , x22 2,
• and the transfer map HnBH → Hn−4BG is the H∗BG-module map taking
x = r0(x) + r1(x)x2 + r2(x)x
2
2 to r2(x), where r ∗i(x) ∈ H BG.
59
APPENDIX A
THE A(1)-ACTION ON H∗BPSp(3)
In this section, let G = PSp(3) be the quotient of Sp(3) by its center ±I (here I
is the identity matrix), and let H = PSp(2, 1) be the quotient of Sp(2) × Sp(1) by
±I. Notice that H is a subgroup of G and the inclusions
Sp(1, 1, 1) → Sp(2, 1) → Sp(3)
induce inclusions
i1 : PSp(1, 1, 1) → H and i2 : H → G
(here PSp(1, 1, 1) := P(Sp(1)3)). Let i : Z42 → PSp(3) be the composition
Z2 × Z2 −j−×→1 PSp(1)× Z2 −∆−×−j−1×2 2 2 −→
j2
PSp(1, 1, 1),
where j maps (1, 0) to i and (0, 1) to j, ∆ is the diagonal, and jn sends each generator
to −1 in the n-th factor. Thus,
(1, 0, 0, 0) 7→ [i, i, i]
(0, 1, 0, 0) →7 [j, j, j]
(0, 0, 1, 0) 7→ [−1, 1, 1]
(0, 0, 0, 1) →7 [1,−1, 1],
etc. For example, (1, 1, 0, 1) 7→ [k,−k, k]. (Note that the images of [i, i, i] and [j, j, j]
indeed commute in PSp(1)3).) Writing Z2 as the multiplicative group ±1, we have
(x1, x2, y1, y2) 7→ [(−1)y1ix1jx2 , (−1)y2ix1jx2 , ix1jx2 ].
Given a compact Lie group K, any representation ρ : K → GLn(R) gives rise to
a real vector bundle over BK which we denote Eρ. Define a four-dimensional real
60
representation Rij of PSp(1, 1, 1) acting on H via [h1, h2, h3] ·x = hixhj. If i = j, this
action fixes the R-span of 1, so we can decompose Rii as the sum of a trivial represen-
tation and a -dimensional representation . The inclusion Z4 × Z2 →−i3 Ri 2 2 PSp(3) then
defines a bundle over B((Z/2)2 × (Z/2)2) ∼= (RP∞)4, and we can identify w(ERij)
with its image in H∗((RP∞)4;Z/2) ∼= Z/2[x1, x2, y1, y2].
We will now compute how ρij transforms the elements xi, yj. In general, we have
ρij(x1, x2, y1, y2)(h) = [(−1)y1ix1jx2 , (−1)y2ix1jx2 , ix1jx2 ] · h.
Case 1: i = j. Then
ρ (x , x , y , y )(h) = [(−1)y1ix1jx2 , (−1)y2ii 1 2 1 2 ix1jx2 , ix1jx2 ] · h
= ix1jx2hix1jx2
= ix1jx2hjx2ix1
= (−1)x1+x2ix1jx2hjx2ix1 .
In particular,
ρii(x1, x2, y1, y2)(1) = (−1)x1+x2(ix1(jx2jx2)ix1) = 1.
When h = i, note that jij = jk = i, so
ρii(x1, x2, y1, y2)(i) = (−1)x1+x2ix1(jx2ijx2)ix1
= (−1)x1+x2(ix1)i(ix1)
= (−1)x1+x2i2x1+1
= (−1)x2i.
Next
ρii(x1, x2, y1, y2)(j) = (−1)x1+x2ix1(jx2jjx2)ix1
61
= (−1)x1+x2(ix1)j2x2+1(ix1)
= (−1)x1ix1jix1
= (−1)x1j
and
ρ (x , x , y , y )(k) = (−1)x1+x2 x1ii 1 2 1 2 i (jx2kjx2)ix1
= (−1)x1+x2(ix1)k(ix1)
= (−1)x1+x2k.
Thus the total Stiefel-Whitney class of i∗(ERii) is thus
Bi∗(w(ERii)) = (1 + x1)(1 + x2)(1 + x1 + x2).
Case 2: i ̸= j. We have
ρ23(x1, x2, y1, y2)(h) = (−1)y2ρ11(x1, x2, y1, y2)(h)
ρ13(x , x , y , y )(h) = (−1)y11 2 1 2 ρ11(x1, x2, y1, y2)(h)
ρ12(x1, x2, y1, y2)(h) = (−1)y1+y2ρ11(x1, x2, y1, y2)(h).
Thus, if we let y3 = y1 + y2,
Bi∗(w(ER23)) = (1 + y2)(1 + x1 + y2)(1 + x2 + y2)(1 + x1 + x2 + y2)
Bi∗(w(ER13)) = (1 + y1)(1 + x1 + y1)(1 + x2 + y1)(1 + x1 + x2 + y1)
Bi∗(w(ER12)) = (1 + y3)(1 + x1 + y3)(1 + x2 + y3)(1 + x1 + x2 + y3).
Next we compute w(BG) and the action by Sq1, Sq2. First,
Bi∗(w(ERii)) = 1 + (x
2
1 + x
2
1x2 + x2) + x1x2(x1 + x2) (mod 2).
62
Let t = x22 1+x1x2+x22 and t3 = x1x2(x +x ) so that w(i∗1 2 ERii) = 1+ t2+ t3. When
i ̸= j, we define as yℓ,
Bi∗(w(ER 2 2 2 2 2 4ij)) = 1 + x1 + x1x2 + x2 + x1x2(x1 + x2) + x1x2(x1 + x2)yℓ + (x1 + x1x2 + x2)yℓ + yℓ
= 1 + t2 + t3 + t3yℓ + t2y
2 4
ℓ + yℓ .
We can write this fourth order term as sk = t3yℓ+t 2 42yℓ +yℓ , where {i, j, k} = {1, 2, 3}.
To summarize,

1 + t2 + t3, if i = jBi∗(w(ERij)) =  (A.1)1 + t2 + t3 + sk, if {i, j, k} = {1, 2, 3}
where
s1 = t3y2 + t2y
2
2 + y
4
2
s2 = t3y1 + t y
2 + y42 1 1
s3 = t3y3 + t
2 4
2y3 + y3.
Remark A.1. There seems to be a minor indexing error in Stolz’s work. He claims
ρ y1 y2 2 423 = (−1) ρ11 and ρ13 = (−1) ρ11. Then sk = t3yk + t2yk + yk, which cleans up
some notation. This essentially amounts to switching the role of y1 and y2 in the map
i. In his version, (0, 0, 1, 0) would map to [1,−1, 1] and (0, 0, 0, 1) maps to [−1, 1, 1].
Next we consider the adjoint representation g of G, which is equivalent to the
conjugation action of G skew-Hermitian 3× 3 quaternionic matrices.
We claim that, restricted to P (Sp(1)3), the representation g decomposes as
R1 ⊕R2 ⊕R3 ⊕R23 ⊕R13 ⊕R12.
The restriction here is from the i2 ◦ i1, where
63
Z 2 × Z 2 →−i( /2) ( /2) P (Sp(1)3) −→i1 iP (Sp(2)× Sp(1)) −→2 PSp(3).
It follows that
B(i i i)∗(w(Eg)) = t32 1 (t+ s1)(t+ s2)(t+ s3),
where t = 1 + t3 + t3. To simplify this, it helps to notice that
s1 + s2 = t3(y1 + y2) + t (y
2 + y22 1 2) + y
4 4
1 + y2
= t3(y1 + y2) + t2(y1 + y )
2
2 + (y1 + y
4
2)
= s3.
We have
B(i2i
∗
1i) (w(Eg)) = (s
2
1s2 + s1s
2
2)t
3 + (s21 + s1s + s
2
2 2)t
4 + t6
and we let t = s2 + s s + s2 and t = s28 1 1 2 2 12 1s2 + s1s22 so that
B(i2i i)
∗
1 (w(Eg)) = t12t
3 + t 48t + t
6. (A.2)
Now we can put this together. From (A.1), we know that t2, t3 are in the image of
Bi∗. We can consider Bi∗ and Bi∗1 2 as bundles with respective fibers HP 1 and HP 2,
and Hurewicz’s theorem shows that Bi∗ and Bi∗1 2 are isomorphisms on cohomology
groups of degree at most 3. In particular, t and t are in the image of Bi i i∗2 3 2 1 . Using
Kono’s computation of H∗BPSp(3), we can thus identify the generators t2 and t3 with
those in H∗BPSp(3).
On the other hand, (A.2) shows t8 and t12 are also in the image of Bi2i ∗1i . These
elements are polynomials in sk, so they are not in the polynomial ring Z/2[t2, t3]. We
can thus identify t8 and t12 with the generators in H∗BPSp(3).
64
A.1 Action by A(1) on BPSp(3)
In the last section, we identified H∗BPSp(3) = Z/(2)[t2, t3, t8, t12] with the subring
of Z2[x1, x2, y1, y2] via
t2 = x
2 2
1 + x1x2 + x2
t3 = x1x2(x1 + x2)
t 2 28 = s1 + s1s2 + s2
t12 = s1s2(s1 + s2)
where
s = t y + t y2 + y41 3 2 2 2 2 and s2 = t3y1 + t y
2 4
2 1 + y1.
For a generator x ∈ H∗BZ2, we have Sq1 x = x2 and Sq2 x = 0. We use this to
compute the actions of Sq1 and Sq2 on cohomology classes.
Theorem A.2. Identifying H∗BPSp(3) = Z2[t2, t3, t8, t12], we have
Sq1 t2 = t Sq
1 t 1 13 3 = 0 Sq t8 = 0 Sq t12 = 0
Sq2 t2 = t
2
2 Sq
2 t3 = t2t3 Sq
2 t8 = 0 Sq
2 t12 = t2t12.
Proof. Using naturality, we compute
Sq1(t ) = Sq12 (x
2
1 + x1x + x
2
2 2)
= Sq(x1)x2 + x1 Sq(x2)
= x2x + x x21 2 1 2
= t3
and
Sq2(t2) = Sq
2(x21 + x1x2 + x
2
2)
65
= x41 + Sq
2(x1)x2 + Sq
1(x1) Sq
1(x2) + x
2 4
1 Sq (x2) + x2
= x41 + x
2x2 + x41 2 2
= (x21 + x1x1 + x
2 2
2)
= t22
(which is expected since t2 has degree 2). Next
Sq1(t ) = Sq13 (x1x2(x1 + x2))
= Sq1(x1x2)(x1 + x2) + x1x2(Sq
1(x1) + Sq
1(x2))
= (x2x + x x21 2 1 2)(x1 + x2) + x1x2(x
2 + x21 2)
= x1x2(x
2 2 2
1 + x2) + x1x2(x1 + x2)
= x1x2(x
2
1 + x
2
2) + x1x
2 2
2(x1 + x2)
= 0
and
Sq2(t 23) = Sq (x1x2(x1 + x2))
= Sq2(x1x2)(x1 + x2) + Sq
1(x x ) Sq11 2 (x1 + x2) + x1x
2
2 Sq (x1 + x2)
= (x1x
2
2) (x1 + x2) + (x
2
1x2 + x1x
2
2)(x
2 2
1 + x2)
= (x1x2)
2(x1 + x2) + (x1x
3
2)(x1 + x2)
= t3(x1x2 + (x1 + x2)
2)
= t2t3.
Continuing,
Sq1(s1) = Sq
1(t3y2 + t y
2 4
2 2 + y2)
= Sq1(t )y + t y23 2 3 2 + Sq
1(t )y22 2
66
= t3y
2
2 + t y
2
3 2
= 0
and similarly Sq1(s2) = 0. Further,
Sq2(s1) = Sq
2(t3y2 + t2y
2
2 + y
4
2)
= Sq2(t )y + t Sq2(y ) + t2y2 + t Sq2(y2) + Sq2(y43 2 3 2 2 2 2 2 2)
= t t y + t2y2 + t y4 + Sq2(y2)y2 + Sq1(y2) Sq1(y2) + y2 2 22 3 2 2 2 2 2 2 2 2 2 2 Sq (y2)
= t t y + t22 3 2 2y
2 4 6 6
2 + t2y2 + y2 + y2
= t2(t3y2 + t y
2 + y42 2 2)
= t2s1
and similarly Sq2(s2) = t2s2. Finally, we can compute
Sq1(t ) = Sq1(s28 1 + s1s2 + s
2
2)
= Sq1(s1)s2 + s1 Sq
1(s2)
= 0
as well as
Sq2(t8) = Sq
2(s21 + s1s2 + s
2
2)
= Sq2(s )s + Sq1(s ) Sq11 1 1 (s1) + s1 Sq
2(s1)
+ Sq2(s 11)s2 + Sq (s1) Sq
1(s2) + s1 Sq
2(s2)
+ Sq2(s2)s + Sq
1
2 (s
1 2
2) Sq (s2) + s2 Sq (s2)
67
= Sq2(s 21)s2 + s1 Sq (s2)
= t2s1s2 + t2s1s2
= 0.
Then
Sq1(t12) = Sq
1(s1s2(s1 + s2))
= Sq1(s1s2)(s1 + s2) + s1s2 Sq
1(s1 + s2)
= 0
and
Sq2(t ) = Sq212 (s1s2(s1 + s2))
= Sq2(s 11s2)(s1 + s2) + Sq (s1s2) Sq
1(s1 + s2) + s1s2 Sq
2(s1 + s2)
= (Sq2(s1)s
1 1
2 + Sq (s1) Sq (s2) + s1 Sq
2(s2))(s1 + s2) + t2s1s2(s1 + s2)
= t2t12.
A.2 Cohomology of BPSp(2, 1)
We write H = PSp(2, 1) := P(Sp(2) × Sp(1)) and G = PSp(3). We have the
bundle
HP 2 = G/H → →−πBH BG.
Let τ denote the corresponding vertical bundle along the fibers of π.
68
Also recall the maps
(Z/2)2 × (Z 2 →−i i i/2) P (Sp(1)3) −→1 H −→2 G.
Note that Bi2 = π.
Claim 2: restricted to H = P (Sp(2)×Sp(1)), the representation g splits as h⊕h⊥,
with h the adjoint representation of H and h⊥ ∼= τ . When restricted to P (Sp(1)3),
h⊥ splits as R13 ⊕R23.
Assuming claim 2 holds,
B(i1i)
∗w(τ) = (t+ s1)(t+ s2)
= t2 + (s1 + s2)t+ s1s2
= 1 + s1 + s2 + t
2
2 + t
2
3 + (s1 + s2)t2 + (s1 + s2)t3 + s1s2.
Define
u2 = t2 u3 = t3 u4 = s1 + s2 u8 = s1s2.
Then
B(i1i)
∗w(τ) = 1 + (u4 + u
2
2) + (u
2
3 + u2u4) + u3u4 + u8.
This means that the class w4(τ) restricts to a nontrivial element in H4HP 2, and
thus the map H∗BH → H∗G/H is surjective. The Serre spectral sequence of π
collapses, and π∗ : H∗BG → H∗BH is thus injective. Moreover, by Leray-Hirsch,
H∗BH is a free H∗BG module with basis {1, w4(τ), w (τ)24 }. We can therefore iden-
tify H∗BH with the subring of Z/(2)[u2, u3, u4, u8] where π∗(t2) = u ∗2, π (t3) = u3,
π∗(t8) = u
2
4 + u8, and π∗(t12) = u4u8. Under this identification, we have
w(τ) = 1 + (u + u2) + (u24 2 3 + u2u4) + u3u4 + u8.
69
APPENDIX B
PRIMITIVE GENERATORS
Since RP∞ is an h-space, the cohomology forms a Hopf algebra. As an algebra,
H∗RP∞ is the polynomial ring Z2[u], where deg(u) = 1. The coalgebra structure is
determined by the fact that the coproduct∆ is an algebra homomo∧rphism and∆(u) =
1⊗ u+ u⊗ 1. For the algebra structure in homology, H ∞∗RP = (x1, x2, x4, x8, . . .)
is the exterior algebra on generators xn, where xn is the linear dual of
n
u2 .
We know that H∗BO = Z2[w1, w2, w3, . . .]. Dually, H∗BO = Z2[u1, u2, u3, . . .],
where u is the linear dual of wii 1. The canonical inclusion j : BO(1) → BO then
satisfies j∗xn = u2n . Clearly there is one indecomposable element in each degree,
hence one primitive in each degree. In this case, the primitive elements are just
sn (the symmetric polynomial generated by xn1 + · · · + xnk written in terms of the
elementary symmetric polynomials wi, where k ≥ n). s
As an algebra, H∗BSO = Z2[wn : n ≥ 2]. There is one indecomposable element
in degree n for every n ≥ 2, so, dually, P nH∗BSO is Z2 for n ≥ 2 and 0 otherwise.
The primitives of H∗BSO are not just those for H∗BO in degrees n ≥ 2. The reason
is because s n+1n is equal to (−1) nwn plus decomposables, so in even degrees we need
to know that sn has a nontrivial decomposable term which is not a multiple of w1.
For example,
s = w4 − 4w24 1 1w2 + 4w1w3 + 2w22 − 4w = w4 ∈ H∗4 1 BO,
so s4 vanishes when restricted to H∗BSO. However, we can use the following lemma.
Lemma B.1. If x is primitive, then nx2 is primitive for all n.
Proof. Using the Frobenius map,
n n n n n
∆(x2 ) = ∆(x)2 = (1⊗ x+ x⊗ 1)2 = 1⊗ x2 + x2 ⊗ 1.
70
For n even, we write n = 2km for some k ≥ 1 and odd m. Then as long as m ≥ 1,
we know sm is a nontrivial primitive in H∗BSO. The primitive in degree n is therefore
k
s2m . We still have not accounted for the primitives whose degrees are powers of 2.
For degree reasons, w2 is primitive (note that w2 ̸= s2 since s2 = s21 (mod 2)). This
means w2n2 is the primitive element of degree 2n+1. We summarize this below.
Theorem B.2. The coalgebra H∗BSO has one primitive element vn of degree n for
all n ≥ 2, where w2, if n = 2
vn = 
s , if n is odd
 ns2n/2, if n > 2 is even.
Proof. We alreadycomputed the following:
sn, if n is odd
v kn = s2 k m
, if n = 2 m, where m > 1 is odd and k ≥ 1
 2k−1w2 , if n = 2k.
If n is even and not a power of 2, we can write n = 2km for k ≥ 1 and m > 1 odd.
We then have kvn = s2m . If k > 1, then n/2 is also even and not a power of 2, so
v = s2
k−1
n/2 m . In this case vn = v2n/2. Now if k = 1 (so n = 2m), then vn = s
2
m and
vn/2 = vm = sm. Again we have v = v2n n/2. Next suppose n = 2
k. For all k ≥ 1, we
saw k−1vn = w22 . When k > 1, n/2 = 2k−1 is also an even power of , so 2
k−2
2 vn/2 = w2 .
Thus vn = v2n/2 as claimed.
71
Theorem B.3. The primitive elements of H∗BSpin comprise one generator zn for
each n ≥ 4 not of the form 2s + 1, where
sn, if n ̸= 2
s + 1 is odd
 n/4w4 , if α(n) = 1 and n is a power of 2zn = 
 n/(2m)sm,m , if α(n) = 2 and n = 2km for m = 2s + 1
z2n/2, if α(n) ≥ 3 and n is even.
Here α(n) is the number of 2-bits in the binary expansion of n. The first definition
only applies for n ̸= 2s + 1, but this covers all odd-degree primitive generators; the
third through fourth definitions together cover the all even degrees starting with 4.
Proof. In H∗BO, sn is nwn plus decomposable elements, so sn restricts to a nontrivial
primitive in H∗BSpin provided that n is odd, n ≥ 4, and n ̸= 2s + 1 (these are the
degrees in which wn is nontrivial). When n is even, start by writing n = 2km for
k ≥ 1 and for odd m > 1. If m ≠ 2s + 1 (note that this precludes m = 3), then sm is
a nontrivial primitive in H∗BSpin and we can set 2kzn = sm . Equivalently, zn = z2n/2.
The condition m ̸= 2s + 1 means α(n) ̸= 2, and m > 1 means α(n) ̸= 1. Finally we
consider when m = 2s + 1. Then n = 2k(2s + 1) = 2k+s + 2k, so α(n) = 2. The class
sm is not primitive, and in fact is zero by lemma B.4. It follows that sm,m is primitive
since
∆(sm,m) = sm,m ⊗ 1 + sm ⊗ sm + 1⊗ sm,m = sm,m ⊗ 1 + 1⊗ sm,m.
We can now define z = s2k−1n m,m for these values of n.
Lemma B.4. For all k, s2k+1 vanishes when restricted to H∗BSpin.
Proof. Stong shows that the natural map H∗BSO → H∗BSpin is epic with kernel
Aw2. In particular w2 vanishes in H∗BSpin, where also w2 = s2. By definition s2k+1
72
∑ k
is the sum 2 +1i xi expressed in terms of the elementary symmetric polynomials wj
in the xi. We can compute the total square easily:(∑ )
k
Sq (s2k+1) = S∑q ( x
2 +1
i )2k+1
= ∑(x(i + x2i ) ( ) )k
= ∑( x + x2 (x + x2
2
( i i ) i i ))k k+1
= ∑ ( xi + x2 2 2i xi + xi )
2k+1 2k+2 2k+1= ( x + x + x +1 2
k+1+2
i i i + xi
= s2k+1 + s2k+2 + s2k+1+1 + s2k+1+2.
k
In particular Sq2 (s2k+1) = s2k+1+1, so by induction s2k+1 vanishes for all k ≥ 0 in
H∗BSpin.
Remark B.5. A potential point of confusion arises when we identify H∗BSpin with
Z k2[wn : n > 2, n ≠ 2 + 1]. Namely, this is an isomorphism of rings, but not of
A-modules, assuming the A-action on Stiefel-Whitney classes is inherited from the
action on H∗BO. It is better to therefore identify H∗BSpin with the quotient of
H∗BSO = Z2[wn : n ≥ 2] by Aw2. To highlight the potential issue, consider the
class s17. Lemma B.4 shows s17 is trivial in H∗BSpin, but direct computation (using
Newton’s identities, for example) shows that z17 does not vanish when simply setting
w = 0 for all n ≥ 2, n = 2kn + 1. In particular s17 = w7w10 + w6w11 + w4w13. This
is a low-degree case which demonstrates H∗BSpin is not isomorphic to Z2[wn : n >
2, n ̸= 2k + 1] as an A-module.
Theorem B.6. We have PHnBSpinc ∼= Z2 whenever n ≥ 2 and n ̸= 2k + 1, and
PHnBSpinc = 0 for other n. Let zn be the primitive generator of PHnBSpinc in
73
degree n for some n ≥ 2 with n ≥2k + 1. Thensn, if α(n) ≥ 3,
zn = s n/2,n/2
, if α(n) = 2,
 n/2w2 , if α(n) = 1.
Proof. The Girard formula shows that sn is nwn plus decomposable elements. Thus
zn = sn for all odd n. If n is a power of 2, then
n/2
zn = w2 . The remaining case is
when n = 2im for odd m > 1 and i > 0. When m ̸= 2k + 1, sm is primitive, so
i
z = s2 = s = s . If m = 2kn m 2im n + 1, one can show that sm,m is primitive, and thus
i−1
zn = s
2
m,m = sn/2,n/2. This proves the claim.
Note that over Z, s2n = 2sn,n + s2n, so sn,n = 1(s2n − s2n). We can use this to2
compute sn,n modulo 2 if we know sn over Z.
74
APPENDIX C
COMPUTATION OF THE Spinc TRANSFER MAP
All coefficients are taken in Z2 unless stated otherwise. In this proof we refer to
integration over the fiber, which we define first here: let X be a Poincaré duality
space of formal dimension n with a (known) orientation, and let πX → E →− B be a
fibration for which π1B acts on X by orientation preserving homotopy equivalences.
Then integration along the fiber π : Hk+nE → Hk! B is defined as the composite
Hk+nE ↠ Ek,n∞ ↪→ E
k,n = Hk2 (B;H
nX) → HkB.
The first two maps come from the fact that Ek,ℓ2 = 0 for ℓ > n and the last map
comes from the orientation.
Theorem C.1. The group G := SU(3) acts transitively on the space CP2 with fiber
H := S(U(1)× U(2)). This gives a bundle
CP2 → πB(S(U(2)× U(1)) →− BSU(3).
with associated vertical bundle τ . We exhibit classes cohomology xi, yi and prove the
following:
1. H∗BH ∼= Z2[x2, x4];
2. H∗BG = Z2[y4, y6];
3. Bπ∗y 24 = x2 + x and Bπ∗4 y6 = x2x4;
4. we have
a) Sq1(y4) = 0 and Sq2(y4) = y6;
b) Sq1(y6) = 0 and Sq2(y6) = 0;
75
5. w(τ) = 1 + x2 + x4
6. π! : HnBH → Hn−4BG isgiven modulo y6 by
 a/2−1y4 , if a > 0 is even and b = 0,
π a b!(x2x4) = y
b−1
4 , if a = 0 and b > 0,
0, if otherwise.
Proof of (1)− (4): We have a commuting diagram as shown, where the horizontal
rows are fiber bundles.
f
U(1) BS(U(1)× U(2)) B(U(1)× U(2))
π j
g
U(1) BSU(3) BU(3)
U(1) EU(1) BU(1)
The map BU(3) → BU(1) is induced by the determinant. Comparing spectral
sequences shows that the generator of H1U(1) transgresses to c2, where H∗BU(3) =
Z ∗2[c2, c4, c6]. Thus H BSU(3) = Z ∗2[y4, y6] where g c4 = y4 and g∗c6 = y6. Next, write
H∗BU(1) = Z2[a2] and H∗BU(2) = Z ∗2[b2, b4]. Using this to identify H B(U(1) ×
U(2)) = Z2[a2, b2, b4], we can compute j∗ via the product formula for Chern classes
as shown.
g∗c2 = 0 j
∗c2 = a2 + b2
g∗c ∗4 = y4 j c4 = a2b2 + b4
g∗c6 = y
∗
6 j c6 = a2b4.
By composing f with the projection to BU(2), we obtain generators x2 = f ∗b2 and
x4 = f
∗b4 in H∗BS(U(1)× U(2)) = Z2[x2, x4]. Comparing spectral sequences again,
76
the generator of H1U(1) in the top row transgresses to j∗c2 = a2 + b2, and this is the
only nontrivial differential. Thus we also have f ∗a2 = f ∗(a2 + b2 + b2) = f ∗b2 = x2.
Finally we compute
π∗y = π∗g∗4 c4 = f
∗j∗c4 = f
∗(a2b
2
2 + b4) = x2 + x4
and
π∗y = π∗g∗6 c6 = f
∗j∗c6 = f
∗(a2b4) = x2x4,
thus establishing parts (1)− (3) of the theorem. For part (4), note that Sq1 y4 =
Sq1 y6 = 0 for dimension reasons. Temporarily write x, y, z for the (mod 2) Chern
roots of BU(3). Then Sq2 c4 = c2c4 + c6 and Sq2 c6 = c2c6 as shown.
Sq2 c4 = Sq
2(xy + xz + yz) Sq2 c6 = Sq
2(xyz)
= (x+ y)xy + (x+ z)xz + (y + z)yz = (x+ y + z)(xyz)
= (x+ y + z)(xy + xz + yz) + 3xyz = c2c6
= c2c4 + c6
Thus Sq2 y = Sq24 g∗(c4) = g∗ Sq2(c4) = g∗(c2c4 + c6) = y6, and similarly Sq2 y6 = 0.
Proof of (5): Let τ̃ be the vertical bundle of B(U(2)×U(1)) → BU(3). The total
space of the universal U(3)-bundle is EU(3) ×U(3) g, where g is the Lie algebra of
U(3). Considering B(U(2)× U(1)) as EU(3)/(U(2)× U(1)), we see that
τ̃ = EU(3)× h⊥
where h⊥ ∼= g/h is a chosen orthogonal complement and h is the Lie algebra of
U(2)× U(1).
77
We now compute the action of U(2)×U(1) on h⊥. An arbitrary element in U(2)
and h⊥ can be written respectively as  
a b 0  0 0 v1
−ub ua 0 and  0 0 v 2 ,
0 0 z −v1 −v2 0
where aa + bb = 1 and uu = zz = 1. We write the former as (P, z) and latter as v.
The adjoint action is then (P, z)v(P, z)−1. As matrices this takes v to  0 0 z(av1 + bv2) 0 0 uz(av − bv )

2 1  .
−z(av1 + bv2) uz(bv1 − av2) 0
At the same time      a b v1  av1 + bv2 =   ,
−ub ua v2 u(av2 − bv1)
and thus the adjoint action of (P, z) on v is v 7→ (P, z) · v = Pvz. Include Z32 →
U(2)×U(1) with the first factor mapping to z and the second two mapping to U(2)
(tobe consistent with earlier notation). We thus compute(−1)j 0 0 
    
 (−1)j 0 v1 (−1)
i+jv1
0 (− i1)k 0  · v =   (−1) =   .0 (−1)k v2 (−1)i+kv2
0 0 (−1)i
Writing H∗BZ32 = Z2[x, y, z], we now compute
w(τ̃) = (1 + x+ y)(1 + x+ z) = 1 + y + z + xy + xz + yz + x2.
Previously we wrote H∗B(U(1) × U(2)) = Z2[a2, b2, b4] and identified a2 = x, b2 =
y + z, and b4 = yz. This shows w(t̃) = 1 + b2 + a2b2 + b + a24 2. Finally, we pull back
78
to BS(U(1)× U(2)) to get w(τ) = 1 + x2 + x2 22 + x4 + x2 = 1 + x2 + x4.
Proof of (6): Previously we saw H∗BH = Z2[x2, x4] and H∗BG = Z2[y4, y6] where
π∗y4 = x
2
2+x4 and π∗y6 = x2x4. We now setH∗CP
2 = Z2[u]/(u3). For degree reasons,
the spectral sequence associated to CP2 → BH → BG is trivial, so we necessarily
have i∗x2 = u. Then i∗x2 = u22 , and since π ◦ i is trivial,
i∗x = i∗(x + x2 + x2) = i∗(π∗ 24 4 2 2 (y4) + x2) = u
2.
Now as a module over H∗BG, H∗BH is free with basis {1, u, u2}. Thus for
any x ∈ H∗BH we can uniquely identify x with r0(x) + r1(x)u + r2(x)u2, where
ri(x) ∈ H∗BG. Integration along the fiber of τ is an H∗BG-module morphism
HnBH → Hn−4BG, so it remains to determine this map on the basis elements
1, u, u2. For degree reasons 1 and u map to 0, and u2 maps to either 1 or 0. Since u2
is already an element on H∗CP2, it is in the image of H4BH → E0,4 → E0,4, so u2∞ 2
maps to 1.
To compute the transfer map we now must write all monomials in x2, x4 in terms
of x2 + x = π∗2 4 (y4) and x2x4 = π∗(y 26) in the basis 1, x2, x2. Note that π! is a
H∗BG-module map, so for y ∈ H∗BG and x ∈ H∗BH, we have π!(π∗(y)x) = yπ!(x).
Since
π (xn+kxn) = π ((x x )nxk) = π (π∗(yn)xk) = yn k! 2 4 ! 2 4 2 ! 6 2 6π!(x2) and π (x
nxn+k! 2 4 ) = y
n
6π!(x
k
4),
we only need to compute π n!(x2 ) and π n!(x4 ). We also observe for n ≥ 3
π (xn) = π (xn−2(x2 + x ) + xn−2x ) = π (xn−2)y + π (xn−3! 2 ! 2 2 4 2 4 ! 2 4 ! 2 )y6
and
π (xn) = π (xn−1(x + x2) + xn−1x2) = π (xn−1! 4 ! 4 4 2 4 2 ! 4 )y4 + π (x
n−3 2
! 4 )y6.
79
It now remains to compute the six base cases π!(xi i2) and π!(x4) for 0 ≤ i ≤ 2. For
degree reasons π!(1) = π!(x2) = 0, and our chosen Leray-Hirsch isomorphism gives
π (x2! 2) = 1. Then π!(x4) = π!(x22 + x4) + π!(x22) = y4π!(1) + 1 = 1, and similarly
π!(x
2
4) = π
2 2
!((x2+x4) )+π (x
4) = π (x2(x2+x ))+π (x2x ) = π (x2! 2 ! 2 2 4 ! 2 4 ! 2)y4+π!(x2)y6 = y4.
From the recurrence relation, it follows that, modulo y6, we have π (x2n n−1! 2 ) = y4 ,
π (x2n+1! 2 ) = 0, and π (xn) = y
n−1
! 4 4 for n > 0. This implies statement (6) and completes
the proof of theorem C.1.
80
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