The string coproduct "knows" Reidemeister/Whitehead torsion

TL;DR

This paper demonstrates that the string coproduct is sensitive to Reidemeister and Whitehead torsion, providing a new tool for understanding homotopy invariants and their relation to torsion in topology.

Contribution

It establishes that the string coproduct is not homotopy invariant and relates it explicitly to Reidemeister and Whitehead torsion, linking algebraic invariants to topological structures.

Findings

Coproducts differ on L(1,7) and L(2,7)

Coproducts can be expressed via Reidemeister torsion

Coproducts transform with Whitehead torsion under homotopy

Abstract

We show that the string coproduct is not homotopy invariant. More precisely, we show that the (reduced) coproducts are different on $L(1,7)$ and $L(2,7)$. Moreover, the coproduct on $L(k,7)$ can be expressed in terms of the Reidemeister torsion and hence transforms with respect to the Whitehead torsion of a homotopy equivalence. The string coproduct can thereby be used to compute the image of the Whitehead torsion under the Dennis trace map.