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Graph complexes and cohomology of configuration spaces

By Erica Minuz
PhD Dissertations
January 2020
Abstract:

We study the generalised configuration space of points in a manifold depending on a graph, originally defined by Eastwood and Huggett. In particular, we examine its cohomology through graph complexes. One of those is the graph complex defined by Baranowsky and Sazdanović, denoted by $\mathcal{C}_{BS}$ that is the $E_1$ page of a spectral sequence converging to the homology of this type of configuration space. We compare $\mathcal{C}_{BS}$ with the graph complex GC defined by Kontsevich by defining a map between them.

In order to compute the rational homotopy type of the classical configuration space, Kriz and Totaro define a commutative differential graded algebra that serves as a rational model for it in the case the manifold is a complex projective variety. We generalise this commutative differential graded algebra by describing the complex $R(\Gamma, A)$, that depends on a graph $\Gamma$ and on a commutative differential graded algebra $A$. We prove that the dual complex of $\mathcal{C}_{BS}$ is quasi equivalent to $R(\Gamma, A)$. In the case $\Gamma$ is a complete graph and $M$ is an even dimensional manifold, $R(\Gamma, A)$ is the commutative differential graded algebra that Idrissi proves to be a real model for the classical configuration space of points in $M$.

Finally, we compute the cohomology of the configuration space dependent on a graph of points in $\mathbb{R}^r$, $r\geq 0$. This is a generalization of the classical computation due to Arnold and Cohen that correspond to the case where the graph is complete. The cohomology of this graph configuration space is the cohomology of the painted little disks operad, that we define as a variation depending on a graph of the classical little disks operads.

Thesis advisor: Marcel Bökstedt
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