The configuration functor of a punctured space

TL;DR

This paper investigates the homology of configuration spaces in punctured spaces with specific CW-complex properties, revealing dependence on fundamental groups and actions of mapping class groups, including counterexamples to previous conjectures.

Contribution

It characterizes the Borel-Moore homology of configuration spaces in punctured spaces via fundamental groups and explores the mapping class group's action, disproving a prior conjecture.

Findings

Homology in degree < n is trivial for these configuration spaces.

Homology depends only on the fundamental group of the punctured space.

An example shows mapping class actions can be trivial on homology but nontrivial on fundamental group quotients.

Abstract

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 $\pi_1(U^*,*)$. We describe this homology group in terms of a presentation of $\pi_1(U^*,*)$. 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 $\pi_1(U^*,*)$ 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 $\pi_1(U^*,*)/ \pi_1(U^*,*)^{(k+1)}$. 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 $\pi_1(U^*,*)/ \pi_1(U^*,*)^{(n+1)}$. The former generalizes a theorem of Bianchi-Miller-Wilson and the latter disproves a conjecture of theirs.