Abstract: In this talk, I will discuss two types of symmetric spaces. The first is the moduli space of trees, introduced by S. Whitehouse for studying $\Gamma$-homology and $\mathbb{E}_{\infty}$-obstruction theory. The second is the complex of not 2-connected graphs, introduced by V. Vassiliev for studying knot invariants. Interestingly, their homology as $\Sigma_n$-modules is isomorphic up to a sign representation of the symmetric group $\Sigma_n$ and a degree shift.
I will demonstrate that this homology isomorphism can be lifted to the topological level. Specifically, I will establish that these two spaces are $\Sigma_n$-equivariantly homotopy equivalent to each other in a suitable context. This project is a joint work with Greg Arone and Guchuan Li.
