KSp-characteristic classes determine Spinh cobordism

Buchanan, Jonathan; McKean, Stephen

doi:10.2140/agt.2026.26.485

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$\mathrm{KSp}$-characteristic classes determine $\mathrm{Spin}^h$ cobordism

Jonathan Buchanan and Stephen McKean

Algebraic & Geometric Topology 26 (2026) 485–551

DOI: 10.2140/agt.2026.26.485

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.

Keywords

connective K-theory, quaternionic spin cobordism

Mathematical Subject Classification

Primary: 19L41

Secondary: 53C27

References

Publication

Received: 20 February 2024

Revised: 2 December 2024

Accepted: 25 March 2025

Published: 11 February 2026

Authors

Jonathan Buchanan
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

Stephen McKean
Department of Mathematics Brigham Young University Provo, UT United States

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Volume 26, issue 2 (2026)