## Representation homology and knot contact homology

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This thesis has four parts. In the first part, we introduce and study representation homology of topological spaces, which is a higher 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 Pirashvili's construction of higher Hochschild homology; in fact, we establish a direct relation between the two theories by proving 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 known homology theories associated with spaces (such as the Pontryagin algebra $ H_*(\Omega X) $, the

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Knutson, Allen