A stable splitting of factorisation homology of generalised surfaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866910774575759360 |
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| author | Kranhold, Florian |
| author_facet | Kranhold, Florian |
| contents | For a manifold $W$ and an $E_d$-algebra $A$, the factorisation homology $\int_W A$ can be seen as a generalisation of the classical configuration space of labelled particles in $W$. It carries an action by the diffeomorphism group $\mathrm{Diff}_\partial(W)$, and for the generalised surfaces $W_{g,1}:=(\#^g S^n\times S^n)\setminus\mathring D{}^{2n}$, we have stabilisation maps among the quotients $\int_{W_{g,1}} A\,/\!/\,\mathrm{Diff}_\partial(W_{g,1})$ which increase the genus $g$. In the case where a highly-connected tangential structure $θ$ is taken into account, we describe its stable homology in terms of the iterated bar construction $\mathrm{B}^{2n}A$ and a tangential Thom spectrum $\mathrm{MT}θ$. We also consider the question of homological stability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_07688 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A stable splitting of factorisation homology of generalised surfaces Kranhold, Florian Algebraic Topology 55R80, 57S05, 57R15, 55P47, 55P48, 18N70 For a manifold $W$ and an $E_d$-algebra $A$, the factorisation homology $\int_W A$ can be seen as a generalisation of the classical configuration space of labelled particles in $W$. It carries an action by the diffeomorphism group $\mathrm{Diff}_\partial(W)$, and for the generalised surfaces $W_{g,1}:=(\#^g S^n\times S^n)\setminus\mathring D{}^{2n}$, we have stabilisation maps among the quotients $\int_{W_{g,1}} A\,/\!/\,\mathrm{Diff}_\partial(W_{g,1})$ which increase the genus $g$. In the case where a highly-connected tangential structure $θ$ is taken into account, we describe its stable homology in terms of the iterated bar construction $\mathrm{B}^{2n}A$ and a tangential Thom spectrum $\mathrm{MT}θ$. We also consider the question of homological stability. |
| title | A stable splitting of factorisation homology of generalised surfaces |
| topic | Algebraic Topology 55R80, 57S05, 57R15, 55P47, 55P48, 18N70 |
| url | https://arxiv.org/abs/2310.07688 |