Espaces de configuration g\'en\'eralis\'es. Espaces topologiques $i$-acycliques. Suites spectrales "basiques"

TL;DR

This paper explicitly computes the symmetric group actions on the cohomology of generalized configuration spaces for $i$-acyclic spaces and extends Church's stability theorem to these families, showing stability and polynomial behavior of Betti numbers.

Contribution

It provides explicit character formulas for symmetric group actions on cohomology of generalized configuration spaces and generalizes Church's representation stability theorem to these spaces.

Findings

Explicit character formulas for $S_m$ acting on cohomology when $X$ is $i$-acyclic.

Stability and polynomiality of Betti numbers for large $m$ in these families.

Families of Betti numbers become constant beyond certain bounds depending on the dimension of $X$.

Abstract

The generalized (ordered) configuration spaces associated to a topological space $X$ are the spaces $\Delta_{\leq\ell}X^{m}:=\{(x_1,\ldots,x_{m})\in X^{m}\mid\#\{x_1,\ldots,x_{m}\}\leq \ell\}$ and $\Delta_{\ell}X^{m}:=\Delta_{\leq\ell}X^{m}\setminus \Delta_{\leq\ell-1}$. They are equipped with the action of the symmetric group $S_m$ permuting coordinates. When $X$ has no interior cohomology (i.e. is $i$-acyclic) we are able to compute explicitly the character formula of $S_m$ acting on the cohomology of these spaces, and if $X$ is furthermore a connected and oriented pseudomanifold of dimension $\geq2$ we generalize Church's representation stability theorem to the case of the families $\{\Delta_{\leq m-a}X^m\}_m$ and $\{\Delta_{\ell-a}X^m\}_m$. We show that, for fixed $a,i\in\mathbb N$, the families of representations $\{ S_m: H ^{i}(\Delta_{?m-a}X^{m})\}_{m}$ are monotone and stationary for $m\geq4i+4a$, if $d_{X}=2$, and for $m\geq2i+4a$, if $d_{X}\geq3$. The corresponding families of characters and Betti numbers are (hence) polynomial and the families of integers $\{\mathop{\rm Betti}_{i}({\Delta_{?m-a}X^{m} / S_m})\}_{m}$ are constant within the same range of integers $m$. We further show that the family $\{\mathop{\rm Betti}_{i}({\Delta_{m}X^{m}/ S_m})\}_{m}$ is constant for $m\geq 2i$, if $d_{X}=2$, and for $m\geq i$, if $d_{X}\geq3$. In particular, complex algebraic varieties whether they are smooth on not verify these generalizations of Church's stability theorems.