Gyration Stability for Products

TL;DR

This paper investigates the concept of gyration stability in Poincaré Duality complexes, establishing conditions under which product complexes maintain gyration stability and providing relevant examples.

Contribution

It proves that the product of two Poincaré Duality complexes is gyration stable if one factor is gyration stable, expanding understanding of stability in these topological structures.

Findings

Product stability is preserved when one factor is gyration stable.

Examples of gyration stable complexes are provided.

Gyration operation relates to surgeries on product complexes.

Abstract

A gyration is an operation on Poincar\'{e} Duality complexes that arises from a certain surgery on the product of a given complex $N$ and a sphere, parametrised by a chosen twisting. Of particular recent interest is the notion of gyration stability; that is, $N$ is gyration stable when all of its gyrations have the same homotopy type, regardless of the twisting used. We prove that a product $N\times M$ of two Poincar\'{e} Duality complexes is gyration stable when one of the product terms is itself gyration stable, and provide some examples of interest.