Smith homomorphisms and Spin$^h$ structures
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917894278873088 |
|---|---|
| author | Debray, Arun Krulewski, Cameron |
| author_facet | Debray, Arun Krulewski, Cameron |
| contents | 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 $Ω_{4k}^{\mathrm{Spin}^c} \otimes\mathbb Z[1/2] \cong Ω_{4k}^{\mathrm{Spin}^h} \otimes\mathbb Z[1/2]$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Smith homomorphisms and Spin$^h$ structures Debray, Arun Krulewski, Cameron Algebraic Topology 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 $Ω_{4k}^{\mathrm{Spin}^c} \otimes\mathbb Z[1/2] \cong Ω_{4k}^{\mathrm{Spin}^h} \otimes\mathbb Z[1/2]$. 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. |
| title | Smith homomorphisms and Spin$^h$ structures |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2406.08237 |