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

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.