On the invariance of the string topology coproduct

TL;DR

This paper investigates the conditions under which the string topology coproduct remains invariant under homotopy equivalences, introducing an obstruction class based on higher homotopy data and the 'fake diagonal' to measure invariance failure.

Contribution

It introduces a new obstruction class that quantifies the failure of invariance of the string topology coproduct under homotopy equivalences, extending to higher dimensional loops.

Findings

Obstruction class determines failure of invariance.

Vanishing of the class indicates invariance.

Applicable to higher dimensional loop coproducts.

Abstract

We give a variant of Naef's formula for the failure of invariance of the string topology coproduct under homotopy equivalences, using an obstruction class build from the higher homotopy data one can associate to a homotopy equivalence as well as the ``fake diagonal''. The vanishing of our obstruction class can be seen as a way to measure a form of smallness for homotopy equivalences. We show that the same obstruction rules the failure of invariance for a generalisation of the coproduct to higher dimensional loops.