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.
Categories: math.AT math.GR math.RT