The configuration functor of a punctured space
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915400607858688 |
|---|---|
| author | Looijenga, Eduard Stavrou, Andreas |
| author_facet | Looijenga, Eduard Stavrou, Andreas |
| contents | Let $U$ be a space whose one point compactification $U^*$ is a CW-complex for which the added point $*$ is the only $0$-cell. We observe that the configuration space $Conf_n(U)$ of $n$ numbered distinct points in $U$ has no closed support homology in degree $<n$ and prove that Borel-Moore homology group $H^{cl}_n(Conf_n(U))$ depends only on the fundamental group $π_1(U^*,*)$. We describe this homology group in terms of a presentation of $π_1(U^*,*)$.
A case of interest is when $U$ is a connected closed oriented surface of positive genus minus a finite nonempty set. Then the mapping class group $Mod(U)$ of $U$ acts on both $π_1(U^*,*)$ and ${H^k}(Conf_n(U){)}\cong H^{cl}_ {2n-k}(Conf_n(U))$ and we prove that its action on the latter is through its action on the nilpotent quotient $π_1(U^*,*)/ π_1(U^*,*)^{(k+1)}$. Furthermore, we give an example of a mapping class of a once punctured closed surface $U$ which acts trivially on ${H^n}(Conf_n(U))$, but not on the nilpotent quotient $π_1(U^*,*)/ π_1(U^*,*)^{(n+1)}$. The former generalizes a theorem of Bianchi-Miller-Wilson and the latter disproves a conjecture of theirs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14366 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The configuration functor of a punctured space Looijenga, Eduard Stavrou, Andreas Geometric Topology Algebraic Topology Let $U$ be a space whose one point compactification $U^*$ is a CW-complex for which the added point $*$ is the only $0$-cell. We observe that the configuration space $Conf_n(U)$ of $n$ numbered distinct points in $U$ has no closed support homology in degree $<n$ and prove that Borel-Moore homology group $H^{cl}_n(Conf_n(U))$ depends only on the fundamental group $π_1(U^*,*)$. We describe this homology group in terms of a presentation of $π_1(U^*,*)$. A case of interest is when $U$ is a connected closed oriented surface of positive genus minus a finite nonempty set. Then the mapping class group $Mod(U)$ of $U$ acts on both $π_1(U^*,*)$ and ${H^k}(Conf_n(U){)}\cong H^{cl}_ {2n-k}(Conf_n(U))$ and we prove that its action on the latter is through its action on the nilpotent quotient $π_1(U^*,*)/ π_1(U^*,*)^{(k+1)}$. Furthermore, we give an example of a mapping class of a once punctured closed surface $U$ which acts trivially on ${H^n}(Conf_n(U))$, but not on the nilpotent quotient $π_1(U^*,*)/ π_1(U^*,*)^{(n+1)}$. The former generalizes a theorem of Bianchi-Miller-Wilson and the latter disproves a conjecture of theirs. |
| title | The configuration functor of a punctured space |
| topic | Geometric Topology Algebraic Topology |
| url | https://arxiv.org/abs/2507.14366 |