PhD Dissertations

January 2020

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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Short URL: http://math.au.dk/publs?publid=1156