Localization genus

TL;DR

This paper introduces the concept of localization genus in homotopy theory, exploring spaces that resemble n-spheres through specific functors like Postnikov sections and connected covers.

Contribution

It generalizes the notion of genus to any homotopical localization functor, extending classical and exotic genus concepts in algebraic topology.

Findings

Defines the Postnikov genus of the n-sphere.

Introduces the localization genus for arbitrary localization functors.

Connects classical and exotic genus notions within a unified framework.

Abstract

Which spaces look like an n-sphere through the eyes of the n-th Postnikov section functor and the n-connected cover functor? The answer is what we call the Postnikov genus of the n-sphere. We define in fact the notion of localization genus for any homotopical localization functor in the sense of Bousfield and Dror Farjoun. This includes exotic genus notions related for example to Neisendorfer localization, or the classical Mislin genus, which corresponds to rationalization.