The homotopy type of the Poincaré cobordism category for surfaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916997954011136 |
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| author | Picot, Azélie |
| author_facet | Picot, Azélie |
| contents | We define a version of the surface cobordism category $\mathrm{Cob}_2^{\mathrm{SG}}(X)$ over a base space $X$ where surfaces are considered up to self homotopy equivalences instead of diffeomorphisms. We prove the induced functor $B\mathrm{Cob}_2^{\mathrm{SG}}(-): \mathcal{S} \rightarrow \mathcal{S}$ is not $1$-excisive. We show its first derivative $\partial_1 B\mathrm{Cob}_2^{\mathrm{SG}}(-)$ in the Goodwillie sense is equivalent to a Thom spectrum over $B\mathrm{haut}_*^+(S^2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_07659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The homotopy type of the Poincaré cobordism category for surfaces Picot, Azélie Algebraic Topology Geometric Topology 57R90 (Primary) 18F50, 57P10 (Secondary) We define a version of the surface cobordism category $\mathrm{Cob}_2^{\mathrm{SG}}(X)$ over a base space $X$ where surfaces are considered up to self homotopy equivalences instead of diffeomorphisms. We prove the induced functor $B\mathrm{Cob}_2^{\mathrm{SG}}(-): \mathcal{S} \rightarrow \mathcal{S}$ is not $1$-excisive. We show its first derivative $\partial_1 B\mathrm{Cob}_2^{\mathrm{SG}}(-)$ in the Goodwillie sense is equivalent to a Thom spectrum over $B\mathrm{haut}_*^+(S^2)$. |
| title | The homotopy type of the Poincaré cobordism category for surfaces |
| topic | Algebraic Topology Geometric Topology 57R90 (Primary) 18F50, 57P10 (Secondary) |
| url | https://arxiv.org/abs/2510.07659 |