## Topology and Geometric Group Theory Seminar

Yuri BerestCornell University
Representation homology of spaces
Tuesday, October 31, 2017 - 1:30pm
Malott 203

Let $G$ be a finite-dimensional affine algebraic group defined over a field $k$ of characteristic 0. For any (discrete) group $\pi$, the set of all representations of $\pi$ in $G$ has a natural structure of an algebraic variety (more precisely, affine k-scheme) called the representation variety $Rep_G(\pi)$. If $X$ is a (based) topological space, the representation variety of its fundamental group $Rep_G[π_1(X)]$ is an important geometric invariant of $X$ that plays a role in many areas of mathematics. In this talk, I will discuss a natural homological extension of this construction, called representation homology, that takes into account a higher homotopy information on spaces and has good functorial properties. The representation homology turns out to be computable (in terms of known invariants) in a number of interesting cases (including simply-connected spaces, Riemann surfaces, link complements, lens spaces, ...), some of which I will examine in detail. Time permitting, I will also explain the relation of representation homology
to other homology theories associated with spaces, such as higher Hochschild homology, $S^1$-equivariant homology of free loop spaces and the (stable) homology of automorphism groups of the free groups $F_n$.