From manifolds to invariants of En-algebras

This thesis is the first step in an investigation of an interesting class of invariants of E_n-algebras which generalize topological Hochschild homology. The main goal of this thesis is to simply give a definition of those invariants.

We define PROPs [special characters omitted], for G a structure group sitting over GL (n, [special characters omitted]). Given a manifold with a (tangential) G-structure, we define functors [special characters omitted] 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, [special characters omitted] is equivalent to the n-little discs PROP, and [special characters omitted][M] is defined for any parallelized n-dimensional manifold M.

The invariant we define for a [special characters omitted]-algebra A is morally defined by a derived coend [special characters omitted] for any n-manifold M with a G-structure.

The case T¹ (A; S¹) 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 Poincaré duality. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)