Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

KSp-characteristic classes determine Spin$^h$ cobordism

Jonathan Buchanan, Stephen McKean

TL;DR

This work constructs a 2-local splitting of the Spin^h cobordism spectrum MSpin^h by introducing KSp- Pontryagin and elephant characteristic classes, and an Atiyah–Bott–Shapiro map φ^h: MSpin^h→KSp. Building on the Anderson–Brown–Peterson framework, the authors decompose MSpin^h into summands indexed by partitions and a set of Z-generated Eilenberg–Mac Lane pieces, and prove isomorphisms on Margolis homology to control cohomology. They then compute Spin^h cobordism groups, deriving ranks via partition numbers and detailing torsion growth, culminating in a criterion: Spin^h cobordism is detected by KSp-characteristic numbers together with Z/2-characteristic data. The results yield a robust, computable picture of Spin^h bordism and its connections to KO/KU/KSp theories, with potential applications to explicit representatives, Pin^h bordism, and Conner–Floyd-type phenomena. Overall, the paper provides an explicit, practical 2-local splitting that advances the understanding of quaternionic spin cobordism and its interactions with K-theoretic invariants.

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.

KSp-characteristic classes determine Spin$^h$ cobordism

TL;DR

This work constructs a 2-local splitting of the Spin^h cobordism spectrum MSpin^h by introducing KSp- Pontryagin and elephant characteristic classes, and an Atiyah–Bott–Shapiro map φ^h: MSpin^h→KSp. Building on the Anderson–Brown–Peterson framework, the authors decompose MSpin^h into summands indexed by partitions and a set of Z-generated Eilenberg–Mac Lane pieces, and prove isomorphisms on Margolis homology to control cohomology. They then compute Spin^h cobordism groups, deriving ranks via partition numbers and detailing torsion growth, culminating in a criterion: Spin^h cobordism is detected by KSp-characteristic numbers together with Z/2-characteristic data. The results yield a robust, computable picture of Spin^h bordism and its connections to KO/KU/KSp theories, with potential applications to explicit representatives, Pin^h bordism, and Conner–Floyd-type phenomena. Overall, the paper provides an explicit, practical 2-local splitting that advances the understanding of quaternionic spin cobordism and its interactions with K-theoretic invariants.

Abstract

A classic result of Anderson, Brown, and Peterson states that the cobordism spectrum MSpin (respectively, MSpin) 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 in terms of Eilenberg--Mac Lane spectra and spectra related to symplectic K-theory. This allows us to determine the Spin cobordism groups systematically. We also prove that two Spin-manifolds are cobordant if and only if their underlying unoriented manifolds are cobordant and their KSp-characteristic numbers agree.
Paper Structure (34 sections, 68 theorems, 187 equations, 5 figures, 4 tables)

This paper contains 34 sections, 68 theorems, 187 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. Quick facts about $\mathrm{KSp}$ and $\mathrm{MSpin}^h$
  4. $\mathrm{KSp}$
  5. $\mathrm{Spin}^h(n)$
  6. $\mathrm{Spin}^h$-cobordism
  7. Atiyah--Bott--Shapiro map
  8. Summary of the Anderson--Brown--Peterson splitting
  9. $\mathrm{KO}$-Pontryagin classes
  10. Proof strategy for splitting $\mathrm{MSpin}^h$
  11. Cohomology of $\mathrm{BSpin}^h$ and $\mathrm{MSpin}^h$
  12. Steenrod modules
  13. $\mathrm{KSp}$-Pontryagin and elephant classes
  14. Module structure for $\mathrm{ksp}\langle n\rangle$
  15. The elephant spectrum
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $F$ be the fiber of the map $\mathrm{ko} \to H\mathbb{Z} / 2\mathbb{Z}$ classifying the non-trivial element of $H^0(\mathrm{ko};\mathbb{Z}/2\mathbb{Z})$. Then there are cohomology classes $Z \subset H^*(\mathrm{MSpin}^h;\mathbb{Z}/2\mathbb{Z})$ and a map of spectra that is a $2$-local equivalence.

Figures (5)

  • Figure 1: The $\mathcal{A}_1$-module $E$ and its namesake
  • Figure 2: The $\mathcal{A}_1$-modules $\mathbin{\hbox{o}rigin=c]{180}{$$}}$ and $C$
  • Figure 3: The $\mathbb{CP}^\infty\to *\to K(\mathbb{Z},3)$ spectral sequence
  • Figure 4: The $K(\mathbb{Z},3)\to\mathrm{BO}\langle 8\rangle\to\mathrm{BSpin}$ spectral sequence
  • Figure 5: Two ways of computing $\mathrm{KSp}$-characteristic numbers

Theorems & Definitions (159)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.7: Freed--Hopkins
  • ...and 149 more