Espaces de configuration généralisés. Espaces topologiques $i$-acycliques. Suites spectrales "basiques"

Abstract

The generalized (ordered) configuration spaces associated to a topological space $X$ are the spaces ${\Delta }_{\le \ell }{X}^m:=\{(x_1,\dots ,x_m)\in X^m\mid \#\{x_1,\dots ,x_m\}\le \ell \}$ and ${\Delta }_{\ell }{X}^m:={\Delta }_{\le \ell }{X}^m\setminus {\Delta }_{\le \ell -1}$. 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 $\ge 2$ we generalize Church's representation stability theorem to the case of the families $\{{\Delta }_{\le m-a}{X}^m{\}}_m$ 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 $m\ge 4i+4a$, if $d_X=2$, and for $m\ge 2i+4a$, if $d_X\ge 3$. The corresponding families of characters and Betti numbers are (hence) polynomial and the families of integers $\{{\mathrm{Betti}}_i({\Delta }_{?m-a}{X}^m/S_m){\}}_m$ are constant within the same range of integers $m$. We further show that the family $\{{\mathrm{Betti}}_i({\Delta }_m{X}^m/S_m){\}}_m$ is constant for $m\ge 2i$, if $d_X=2$, and for $m\ge i$, if $d_X\ge 3$. In particular, complex algebraic varieties whether they are smooth on not verify these generalizations of Church's stability theorems.

195 pages, in French. 15 figures