Smith homomorphisms and Spin$^h$ structures
Arun Debray, Cameron Krulewski
TL;DR
This work resolves two questions on Spin^h bordism by constructing a Smith isomorphism between the reduced Spin^h bordism of RP^∞ and Pin^{h-} bordism, and by giving a geometric, rational bridge between Spin^c and Spin^h bordism via an induced p_* map that becomes an isomorphism after inverting 2. The authors develop a general framework of twisted spin structures and Smith homomorphisms, encode them in Smith long exact sequences, and provide explicit geometric interpretations of the maps involved, enabling computable long exact sequences from data (X,V,W). They prove the RP^∞–Pin^{h-} Smith isomorphism using Thom spectrum decompositions and relative Thom isomorphisms, and show that Ω^{Spin^c}_{4k} ⊗ Z[1/2] ≅ Ω^{Spin^h}_{4k} ⊗ Z[1/2], explaining rank coincidences and offering a method to generate Spin^h rational generators from Spin^c generators while clarifying integrality obstructions from 2-torsion. Together, these results advance computability in Spin^h bordism and clarify the relationships among Spin^c, Spin^h, and Pin^h structures with concrete geometric and homotopical tools.
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 $Ω_{4k}^{\mathrm{Spin}^c} \otimes\mathbb Z[1/2] \cong Ω_{4k}^{\mathrm{Spin}^h} \otimes\mathbb Z[1/2]$. 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.