Smith homomorphisms and Spin${}^h$ structures

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 ${\mathbb{R}\mathbb{P}}^{\mathrm{\infty }}$ and pin${}^{h-}$ bordism, and we provide a geometric explanation for the isomorphism ${\mathrm{\Omega }}_{4k}^{{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c}\otimes \mathbb{Z}[1\mathrm{/}2]\cong {\mathrm{\Omega }}_{4k}^{{\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^h}\otimes \mathbb{Z}[1\mathrm{/}2]$. 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.