Year: 2024
Type: article
Abstract: In this article, we answer two questions of Buchanan–McKean [ KSp-characteristic classes determine Spinh cobordism , https://arxiv.org/ abs/2312.08209, 2023] about bordism for manifolds with s p i n h \mathrm {spin}^h structures: we establish a Smith isomorphism between the reduced s p i n h \mathrm {spin}^{h} bordism of R P ∞ \mathbb {RP}^\infty and p i n h − \mathrm {pin}^{h-} bordism, and we provide a geometric explanation for the isomorphism Ω 4 k S p i n c ⊗ Z [ 1 / 2 ] ≅ Ω 4 k S p i n h ⊗ Z [ 1 / 2 ] \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 joint with Devalapurkar, Liu, Pacheco-Tallaj, and Thorngren [ The Smith fiber sequence and invertible field theories , https://arxiv.org/abs/2405.04649, 2024], specifically that the Smith homomorphism participates in a long exact sequence with explicit, computable terms.
DOI: 10.1090/proc/17041
Authors: Arun Debray, Cameron Krulewski
Institutions: University of Kentucky, Massachusetts Institute of Technology
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W4405424642