On the String Topology Coproduct for Lie Groups

TL;DR

This paper investigates the structure of the string topology coproduct on the free loop space of Lie groups, revealing its decomposition and triviality in certain cases, with implications for understanding loop space algebraic structures.

Contribution

It demonstrates that the coproduct splits into a diagonal and a based coproduct, and shows triviality for even-dimensional and many simply connected Lie groups.

Findings

Coproduct splits into diagonal and based coproducts.

Trivial coproduct for even-dimensional Lie groups.

Trivial coproduct for many simply connected Lie groups.

Abstract

The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space. We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional Lie groups. Using results by Bott and Samelson, we show that the coproduct is trivial as well for a large family of simply connected Lie groups.