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Louis Hainaut: Some computations of compact support cohomology of configuration spaces

Time: Fri 2022-05-13 10.00

Location: Kräftriket, House 6, Room 306

Doctoral student: Louis Hainaut

Opponent: Ben Knudsen (Northeastern University)

Supervisor: Dan Petersen

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This licentiate thesis consists of two papers related to configuration spaces of points. In paper I a general formula for the Euler characteristic of configuration spaces on any topologically stratified space X is obtained in terms of geometric and combinatorial data about the strata. More generally this paper provides a formula for the Euler characteristic of the cohomology with compact support of these configuration spaces with coefficients in a constructible complex of sheaves K on X. The formula for the classical Euler characteristic is then obtained by taking K to be the dualizing complex of X. This formula generalizes similar results about configuration spaces on a manifold or on a simplicial complex, as well as another formula for any Hausdorff space X when the complex of sheaves K is trivial. In paper II we study the cohomology with compact support of config- uration spaces on a wedge sum of spheres X, with rational coefficients. We prove that these cohomology groups are the coefficients of an ana- lytic functor computing the Hochschild–Pirashvili homology of X with certain coefficients. Moreover, we prove that, up to a filtration, these same cohomology groups are a polynomial functor in the reduced cohomology of X, with coefficients not depending on X. Contrasting the information provided by two different models we are able to partially com- pute these coefficients, and in particular we obtained a complete answer for configurations of at most 10 points. The coefficients thus obtained can be used to compute the weight 0 part of the cohomology with compact support of the moduli space \(M_{2,n}\).

Read the thesis on DiVA