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arXiv · 2312.08209

KSp-characteristic classes determine Spin$^h$ cobordism

Jonathan Buchanan·Stephen McKean
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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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Jonathan Buchanan, Stephen McKean. 2023-12-13. KSp-characteristic classes determine Spin$^h$ cobordism. https://doi.org/10.2140/agt.2026.26.485

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