Integral String Lie Algebra Structure of Spheres

TL;DR

This paper computes the integral string homology and string bracket for spheres, revealing non-trivial Lie algebra structures on torsion elements, contrasting with prior rational coefficient results.

Contribution

It provides the first integral coefficient computations of string homology and brackets for spheres, showing non-zero brackets on torsion elements.

Findings

String homology for spheres over integers computed.

String bracket is non-zero on torsion elements.

Contrasts with rational coefficient results where brackets are trivial.

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 Somnath Basu computed the string homology and string bracket for spheres over rational coefficients and found that the bracket is trivial in his dissertation. 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 for our computations. We find that over the integers, the string Lie algebra bracket structure is not always zero as Basu found. The string bracket turns out to be non-zero on torsion elements coming from string homology.