Smith homomorphisms and Spin${}^h$ structures

Abstract

In this article, we answer two questions of Buchanan-McKean (arXiv:2312.08209) about bordism for manifolds with spin${}^h$ structures: we establish a Smith isomorphism between the reduced spin${}^h$ bordism of ${\mathbb{RP}}^{\infty }$ and pin${}^{h-}$ bordism, and we provide a geometric explanation for the isomorphism ${\Omega }_{4k}^{{\mathrm{Spin}}^c}\otimes \mathbb{Z}[1/2]\cong {\Omega }_{4k}^{{\mathrm{Spin}}^h}\otimes \mathbb{Z}[1/2]$. Our proofs use the general theory of twisted spin structures and Smith homomorphisms that we developed in arXiv:2405.04649 joint with Devalapurkar, Liu, Pacheco-Tallaj, and Thorngren, specifically that the Smith homomorphism participates in a long exact sequence with explicit, computable terms.

15 pages; comments welcome!

References in corpus (10)

• Reflection positivity and invertible topological phases
• A New SU(2) Anomaly
• Quaternionic Monopoles
• A Non-Perturbative Definition of the Standard Models
• The algebra of anomaly interplay
• Higher Anomalies, Higher Symmetries, and Cobordisms II: Lorentz Symmetry Extension and Enriched Bosonic/Fermionic Quantum Gauge Theory
• What bordism-theoretic anomaly cancellation can do for U
• Interacting SPT phases are not Morita invariant
• Spin${}^h$ Manifolds
• Invariants of real vector bundles

Cited by in corpus (2)

• The Smith Fiber Sequence and Invertible Field Theories
• KSp-characteristic classes determine Spin${}^h$ cobordism