Chain-level equivariant string topology: algebra versus analysis

TL;DR

This paper demonstrates that on 2-connected closed oriented manifolds, the analytic and algebraic methods for constructing an IBL$_ty$ structure are equivalent, preserving important algebraic properties and invariances.

Contribution

It establishes the equivalence of analytic and algebraic constructions of IBL$_ty$ structures on certain manifolds, linking topology, algebra, and analysis.

Findings

Analytic and algebraic IBL$_ty$ structures coincide on 2-connected closed oriented manifolds.

The structure is invariant under orientation-preserving homotopy equivalences.

Induces the involutive Lie bialgebra structure of Chas and Sullivan on homology.

Abstract

We prove that on 2-connected closed oriented manifolds, the analytic and algebraic constructions of an IBL$_\infty$ structure associated to a closed oriented manifold coincide. The corresponding structure is invariant under orientation preserving homotopy equivalences and induces on homology the involutive Lie bialgebra structure of Chas and Sullivan.