Homotopy Non-Invariance of the String Cobracket and the Failure of the Lie Bialgebra Structure

TL;DR

The paper demonstrates that the string cobracket is not a homotopy invariant and that the expected Lie bialgebra structure fails in higher-dimensional string topology.

Contribution

It provides explicit examples showing the non-invariance of the string cobracket and the breakdown of Lie bialgebra structures in higher dimensions.

Findings

String cobracket differs on lens spaces $L(9;1)$ and $L(9;4)$

String cobracket is not a homotopy invariant

The Lie bialgebra structure does not hold for the $S^1$-equivariant homology of the free loop space.

Abstract

We prove that the string cobracket is not a homotopy invariant. Adapting Naef's method arXiv:2106.11307 for computing the string coproduct, we show that the string cobrackets on the three-dimensional lens spaces $L(9;1)$ and $L(9;4)$ differ. We further relate the string cobracket to the Whitehead torsion, analogously to the case of the string coproduct. In addition, we show that the string bracket and the string cobracket do not endow the $S^1$-equivariant homology of the free loop space with a Lie bialgebra structure. These findings indicate that the analogy with the Turaev cobracket breaks down in higher-dimensional string topology.