KSp-characteristic classes determine Spin^h cobordism

2023/12/13 by Buchanan, Jonathan, McKean, Stephen
#Algebraic Topology (math.AT) #FOS: Mathematics #Primary: 19L41. Secondary: 53C27

paper · doi:10.48550/arxiv.2312.08209

Abstract

A classic result of Anderson, Brown, and Peterson states that the cobordism spectrum MSpin (respectively, MSpin^c) splits as a sum of Eilenberg--Mac Lane spectra and connective covers of real K-theory (respectively, complex K-theory) at 2. We develop a theory of symplectic K-theory classes and use these to build an explicit splitting for MSpin^h in terms of Eilenberg--Mac Lane spectra and spectra related to symplectic K-theory. This allows us to determine the Spin^h cobordism groups systematically. We also prove that two Spin^h-manifolds are cobordant if and only if their underlying unoriented manifolds are cobordant and their KSp-characteristic numbers agree.

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