String Homology and Lie Algebra Structures (Ph.D. Thesis)

TL;DR

This thesis computes string homology and the string bracket for spheres with integer coefficients, revealing non-trivial Lie algebra structures and torsion effects, extending previous rational and mod 2 results.

Contribution

It provides the first detailed computation of string homology and brackets for spheres over integers, highlighting torsion contributions and differences from rational cases.

Findings

String bracket is non-zero on torsion elements.

Computed string homology for spheres with integer coefficients.

Analyzed string Lie algebra structures of surfaces.

Abstract

Chas and Sullivan introduced string homology, which is the equivariant homology of the loop space with the $S^1$ action on loops by rotation. Craig Westerland computed the string homology for spheres with coefficients in $\mathbb{Z} /2\mathbb{Z}$, and in Somnath Basu's dissertation, he computes the string homology and string bracket for spheres over rational coefficients, and he finds that the bracket is trivial. In this paper, we compute string homology and the string bracket for spheres with integer coefficients, treating the odd- and even-dimensional cases separately. We use the Gysin sequence and Leray-Serre spectral sequence to aid in our computations. We find that over the integers, the string Lie algebra bracket structure is more interesting, and not always zero. The string bracket turns out to be non-zero on torsion coming from string homology. We also make some computations of the Goldman Lie algebra structure, and more generally, the string Lie algebra structure of closed, orientable surfaces.