Twisted homology of configuration spaces and homology of spaces of equivariant maps
For a positive integer $N$ and a topological space $X$, let $I(X,N)$ denote the ordered configuration space of $N$ distinct points in $X$ defined by $I(X,N)=\{(x_1,\dots ,x_N)\in X^N:x_i\not= x_j\text{ if }i\not= j\}$. The symmetric group $S(N)$ of $N$-letters acts freely on $I(X,N)$ by permutation of coordinates and let $B(X,N)$ denote the unordered configuration space of $N$ distinct points in $X$ given by the orbit space $B(X,N)=I(X,N)/S(N)$. For a local system $L$ of groups on $X$, let $\overline{H}_*(X;L)$ denote the (Borel-Moore) homology group of $X$ with coefficients $L$. \par Let us consider the non-trivial local system of groups on the $m$ dimensional real projective space $\mathbb{R}\mathrm{P}^m$; the non-zero element of $\pi_1(\mathbb{R}\mathrm{P}^m)$ acts on the fibers as multiplication by $-1$. Let $\Theta$ denote the local system on the space $(\mathbb{R}\mathrm{P}^m)^N$ given by the tensor product of $n$ systems lifted by the standard projection from the above systems on the factors. \par Let $\pm \mathbb{Q}$ (resp. $\tilde{\Theta}$) denote the local system on $B(\mathbb{R}\mathrm{P}^m,N)$ with fiber $\mathbb{Q}$, which resolves over the loops defining odd permutations of $N$ points (resp. the loops such that the union of traces of all $N$ points during the movement along the loop defines the non-zero element of the group $H_1(\mathbb{R}\mathrm{P}^m;\mathbb{Z}/2)$). Note that the restriction of $\Theta$ to the subspace $I(\mathbb{R}\mathrm{P}^m,N)$ is isomorphic to the local system lifted from $\tilde{\Theta}$ by the standard covering $I(\mathbb{R}\mathrm{P}^m,N)\to B(\mathbb{R}\mathrm{P}^m,N)$. \par In this paper the author investigates the homology groups $\overline{H}_*(B(\mathbb{R}\mathrm{P}^m,N);L)$ $(L=\tilde{\Theta}$ or $\tilde{\Theta}\otimes \pm \mathbb{Q}$) and $H_*(I(\mathbb{R}\mathrm{P}^m,N);\Theta)$. In particular, when $m$ is odd, he computes the Poincaré series of the group $H_*(B(\mathbb{R}\mathrm{P}^m,N);L)$ explicitly and he shows that the group $H_*(I(\mathbb{R}\mathrm{P}^m,N);\Theta)$ is trivial if $N$ is odd. Moreover, when $m<M$ are odd numbers, he also studies the space of equivariant maps $f:S^m\to S^M$ and he determines the Poincaré series of the homology of this space with rational coefficients explicitly. Finally, he considers the space $\Phi (\mathbb{R}\mathrm{P}^m,N)$ given by the fiber of the first projection $I(\mathbb{R}\mathrm{P}^m,N)\to \mathbb{R}\mathrm{P}^m$ and he computes the Poincaré polynomial of the homology $H_*(\Phi (\mathbb{R}\mathrm{P}^m,N);\Theta)$ explicitly when $N$ is even and $m$ is odd. His proofs are based on a careful analysis of the Vassiliev spectral sequence (invented by him) constructed by using simplicial resolutions.
scientific article; zbMATH DE number 7036541