Simple homotopy invariance of the loop coproduct

TL;DR

This paper establishes that the Goresky-Hingston loop coproduct in string topology remains invariant under simple homotopy equivalences of manifolds, linking it to Whitehead torsion and framed configuration spaces.

Contribution

It provides a transformation formula involving Whitehead torsion, proving the loop coproduct's invariance under simple homotopy equivalences and connecting it to TQFT structures.

Findings

Loop coproduct is invariant under simple homotopy equivalences.

Transformation formula involves Whitehead torsion.

Loop coproduct arises as a secondary operation in 2D TQFT.

Abstract

We prove a transformation formula for the Goresky-Hingston loop coproduct in string topology under homotopy equivalences of manifolds. The formula involves the trace of the Whitehead torsion of the homotopy equivalence. In particular, it implies that the loop coproduct is invariant under simple homotopy equivalences. In a sense, our results determine the Dennis trace of the simple homotopy type of a closed manifold from its framed configuration spaces of $\leq 2$ points. We also explain how the loop coproduct arises as a secondary operation in a 2-dimensional TQFT which elucidates a topological origin of the transformation formula.