Smith homomorphisms and mjchᵇ structures

In this article, we answer two questions of Buchanan–McKean [ KSp-characteristic classes determine Spinh cobordism , https://arxiv.org/ abs/2312.08209, 2023] about bordism for manifolds with s p i n h spin^h structures: we establish a Smith isomorphism between the reduced s p i n h spinʰ bordism of R P ∞ RP^∞ and p i n h − pinʰ⁻ bordism, and we provide a geometric explanation for the isomorphism Ω 4 k S p i n c ⊗ Z [ 1 / 2 ] ≅ Ω 4 k S p i n h ⊗ Z [ 1 / 2 ] Ω ₄ₖSpin^c⊗ Z[1/2] Ω ₄ₖSpin^h⊗ Z[1/2] . Our proofs use the general theory of twisted spin structures and Smith homomorphisms that we developed joint with Devalapurkar, Liu, Pacheco-Tallaj, and Thorngren [ The Smith fiber sequence and invertible field theories , https://arxiv.org/abs/2405.04649, 2024], specifically that the Smith homomorphism participates in a long exact sequence with explicit, computable terms.