Papers › Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting
Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting
Weiyan Chen
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We consider two families of algebraic varieties Yₙ indexed by natural numbers n: the configuration space of unordered n-tuples of distinct points on ℂ, and the space of unordered n-tuples of linearly independent lines in ℂⁿ. Let Wₙ be any sequence of virtual Sₙ-representations given by a character polynomial, we compute Hⁱ(Yₙ; Wₙ) 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 lim_(n→∞)Hⁱ(Yₙ; Wₙ) 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.
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