On genera of polyhedra

TL;DR

This paper investigates the classification of polyhedra in the stable homotopy category by their genera, providing criteria for when two polyhedra belong to the same genus and exploring examples of such classifications.

Contribution

It establishes a new criterion for determining when two polyhedra are in the same genus based on stable isomorphisms after localization, and explores properties of these genera.

Findings

Y is in the genus of X if and only if XvB is stably isomorphic to YvB.

If XvX and XvY are stably isomorphic, then X and Y are stably isomorphic.

Several examples of genus calculations are provided.

Abstract

We consider genera of polyhedra (finite cell complexes) in the stable homotopy category. Namely, the genus of a polyhedron X is the class of polyhedra Y such that all localizations of Y are stably isomorphic to the corresponding localizations of X. We prove that Y is in the genus of X if and only if the wedge XvB is stably isomorphic to YvB, where B is the wedge of all spheres S^n such that the n-th stable homotopy group of X is not torsion. We also prove that if XvX and XvY are stably isomorphic, so are also X and Y. Several examples of calculations of genera are considered.