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Tuesday, April 28, 2015

**Abstract:** We describe how some classical theorems on the Hochschild homology of cochains in X lift to theorems about the topological Hochschild homology of the ring spectrum DX. In particular, we focus on the duality between THH(DX) and LX, which is a special case of the Ayala-Francis Poincare/Koszul duality theorem. When X is a smooth manifold, this duality may be interpreted as Atiyah duality for the free loop space LM, which is an infinite-dimensional manifold. We also explain how to prove that this duality is equivariant, using some surprising technical results that have consequences for the general study of orthogonal G-spectra.