Scalar Curvature and Transfer Maps in Spin and Spin^c Bordism
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Date
2024-01-10
Authors
Granath, Elliot
Journal Title
Journal ISSN
Volume Title
Publisher
University of Oregon
Abstract
In 1992, Stolz proved that, among simply connected Spin-manifolds of dimension5 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. In this paper we prove an analogous result for Spinc- ∗−8 ∗
manifolds and a related invariant αc : ΩSpinc → ku. We show that ker αc is the ∗
sum of the image of Stolz’s transfer ΩSpinBPSp(3) → ΩSpinc and an analogous map ∗−8 ∗
ΩSpinc BSU(3) → ΩSpinc . Finally, we expand on some details in Stolz’s original paper ∗−4 ∗
and provide alternate proofs for some parts.
Description
Keywords
bordism, bundle, cobordism, spin, topology, transfer