## All Dissertations, Theses, and Capstone Projects

9-2015

Dissertation

Ph.D.

#### Program

Mathematics

Dennis P. Sullivan

Mathematics

#### Keywords

hochschild; integrals; iterated; string; topology

#### Abstract

The field of string topology is concerned with the algebraic structure of spaces of paths and loops on a manifold. It was born with Chas and Sullivan's observation of the fact that the intersection product on the homology of a smooth manifold $M$ can be combined with the concatenation product on the homology of the based loop space on $M$ to obtain a new product on the homology of $LM$, the space of free loops on $M$. Since then, a vast family of operations on the homology of $LM$ have been discovered.

In this thesis we focus our attention on a non trivial coproduct of degree $1-\text{dim}(M)$ on the homology of $LM$ modulo constant loops. This coproduct was described by Sullivan on chains on general position and by Goresky and Hingston in a Morse theory context. We give a Thom-Pontryagin type description for the coproduct. Using this description we show that the resulting coalgebra is an invariant on the oriented homotopy type of the underlying manifold. The coproduct together with the loop product induce an involutive Lie bialgebra structure on the $S^1$-equivariant homology of $LM$ modulo constant loops. It follows from our argument that this structure is an oriented homotopy invariant as well.

There is also an algebraic theory of string topology which is concerned with the structure of Hochschild complexes of DG Frobenius algebras and their homotopy versions. We make several observations about the algebraic theory around products, coproducts and their compatibilities. In particular, we describe a $BV$-coalgebra structure on the coHochschild complex of a DG cocommutative Frobenius coalgebra. Some conjectures and partial results regarding homotopy versions of this structure are discussed.

Finally, we explain how Poincaré duality may be incorporated into Chen's theory of iterated integrals to relate the geometrically constructed string topology operations to algebraic structures on Hochschild complexes.

COinS