Smith homomorphisms and Spinʰ structures

In this article, we answer two questions of Buchanan-McKean (arXiv:2312.08209) about bordism for manifolds with spinʰ structures: we establish a Smith isomorphism between the reduced spinʰ bordism of RP^∞ and pinʰ⁻ bordism, and we provide a geometric explanation for the isomorphism Ω₄ₖSpinᶜ ⊗ Z[1/2] Ω₄ₖSpinʰ ⊗ 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.