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 pin^h- bordism, and we provide a geometric explanation for the isomorphism ₄₊^Spinᶜ Z1/2 ₄₊^Spinʰ Z1/2. 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.