String topology in three flavours

TL;DR

This paper explores two key string topology operations, the product and coproduct, from geometric and algebraic viewpoints, providing computations, invariance properties, and establishing their equivalence in certain cases.

Contribution

It introduces geometric and algebraic frameworks for string topology operations, computes examples on lens spaces, and demonstrates their equivalence for simply connected manifolds with real coefficients.

Findings

Coproduct distinguishes 3D lens spaces

Operations are equivalent under rational homotopy for simply connected manifolds

Invariance properties are established for both geometric and algebraic perspectives

Abstract

We describe two major string topology operations, the Chas-Sullivan product and the Goresky-Hingston coproduct, from geometric and algebraic perspectives. The geometric construction uses Thom-Pontrjagin intersection theory while the algebraic construction is phrased in terms of Hochschild homology. We give computations of products and coproducts on lens spaces via geometric intersection, and deduce that the coproduct distinguishes 3-dimensional lens spaces. Algebraically, we describe the structure these operations define together on the Tate-Hochschild complex. We use rational homotopy theory methods to sketch the equivalence between the geometric and algebraic definitions for simply connected manifolds and real coefficients, emphasizing the role of configuration spaces. Finally, we study invariance properties of the operations, both algebraically and geometrically.