Representation homology of spaces and higher Hochschild homology

In this paper, we study representation homology of topological spaces, that is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology in terms of classical (abelian) homological algebra. Our construction is parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by showing that the representation homology of the (reduced) suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other standard homology theories associated with spaces (such as Pontryagin algebras, S^1-equivariant homology of the free loop space and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces and some 3-dimensional manifolds (such as link complements in R^3 and the lens spaces L(p,q)). One of our main results, which we call the Comparison Theorem, expresses the representation homology of a simply-connected topological space of finite rational type in terms of its Quillen and Sullivan models.