On the string topology of symmetric spaces of higher rank

TL;DR

This paper investigates the algebraic structures in the homology of loop spaces of compact symmetric spaces, revealing non-trivial products and trivial coproducts in higher rank cases, with implications for string topology.

Contribution

It demonstrates the non-triviality of the Chas-Sullivan product and the triviality of the coproduct in higher rank symmetric spaces, using explicit cycle constructions.

Findings

Chas-Sullivan product is highly non-trivial for all ranks.

Many non-nilpotent classes correspond to iterated closed geodesics.

Based string topology coproduct is trivial in higher rank symmetric spaces.

Abstract

The homology of the free and the based loop space of a compact globally symmetric space can be studied through explicit cycles. We use cycles constructed by Bott and Samelson and by Ziller to study the string topology coproduct and the Chas-Sullivan product on compact symmetric spaces. We show that the Chas-Sullivan product for compact symmetric spaces is highly non-trivial for any rank and we prove that there are many non-nilpotent classes whose powers correspond to the iteration of closed geodesics. Moreover, we show that the based string topology coproduct is trivial for compact symmetric spaces of higher rank and we study the implications of this result for the string topology coproduct on the free loop space.