From manifolds to invariants of En̳-algebras

Abstract

This thesis is the first step in an investigation of an interesting class of invariants of En-algebras which generalize topological Hochschild homology. The main goal of this thesis is to simply give a definition of those invariants. We define PROPs EG, for G a structure group sitting over GL(n, R). Given a manifold with a (tangential) G-structure, we define functors EG[M]: (EG) 0 -+ Top constructed out of spaces of G-augmented embeddings of disjoint unions of euclidean spaces into M. These spaces are modifications to the usual spaces of embeddings of manifolds. Taking G - 1, El is equivalent to the n-little discs PROP, and El [M] is defined for any parallelized n-dimensional manifold M. The invariant we define for a Es-algebra A is morally defined by a derived coend TG(A; M) := EG[M] 9 A n EG for any n-manifold M with a G-structure. The case T' (A; Sl) recovers the topological Hochschild homology of an associative ring spectrum A. These invariants also appear in the work of Jacob Lurie and Paolo Salvatore, where they are involved in a sort of non-abelian Poincare duality.