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 $\mathbb{RP}^\infty$ 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.