Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting

arXiv1603.03931MaRDI QIDQ6271401

Weiyan Chen

Publication date: 12 March 2016

Abstract: We consider two families of algebraic varieties $Y_n$ indexed by natural numbers $n$: the configuration space of unordered $n$-tuples of distinct points on $mathbbC$, and the space of unordered $n$-tuples of linearly independent lines in $mathbb{C}^n$. Let $W_n$ be any sequence of virtual $S_n$-representations given by a character polynomial, we compute $H^i(Y_n;W_n)$ for all $i$ and all $n$ in terms of double generating functions. One consequence of the computation is a new recurrence phenomenon: the stable twisted Betti numbers $li{m}_{noinfty}dim{H}^i(Y_n;W_n)$ are linearly recurrent in $i$. Our method is to compute twisted point-counts on the $F_q$-points of certain algebraic varieties, and then pass through the Grothendieck-Lefschetz fixed point formula to prove results in topology. We also generalize a result of Church-Ellenberg-Farb about the configuration spaces of the affine line to those of a general smooth variety.

Has companion code repository: https://github.com/LimeHero/RepresentationStability

This page was built for publication: Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6271401)