The $n$-th ordered configuration space of a graph parametrizes ways of placing $n$ distinct and labelled particles on that graph. The homology of the one-point compactification of such configuration space is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We give a cellular decomposition of these configuration spaces on which the actions are realized cellularly and thus construct an efficient free resolution for their homology representations. Using the Peter-Weyl Theorem for symmetric groups, we consider each irreducible $S_n$-representation individually, vastly simplifying the calculation of these homology representations from the free resolution. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$ acting by permuting point labels for all $n\leq 10$. This is joint work with Christin Bibby, Melody Chan, and Nir Gadish. Our paper can be found on arXiv with ID 2109.03302.
The talk will start with a pre-seminar at 2pm:
Title: Tropical moduli spaces
Abstract: I will discuss in more detail the construction of tropical moduli spaces and their connections to configuration spaces of graphs. I will also explain why there exists an $\mathrm{Out}(F_g)$-action on the homology of the compactified configuration spaces of a graph.
Zoom: https://washington.zoom.us/j/689897930
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