Poset-stratified space structures of homotopy sets

TL;DR

This paper introduces a novel way to impose poset-stratified space structures on homotopy sets, capturing their dependence relations and invariants through order-theoretic and cohomological frameworks.

Contribution

It establishes a canonical poset-preorder on homotopy sets and demonstrates how various topological and algebraic invariants induce poset-stratified structures.

Findings

Homotopy sets can be structured as poset-stratified spaces capturing dependence.

Cohomology classes induce similar poset structures reflecting dependence.

Invariants like Gottlieb groups and LS-category also give rise to poset-stratified structures.

Abstract

A poset-stratified space is a pair $(S, S \xrightarrow \pi P)$ of a topological space $S$ and a continuous map $\pi: S \to P$ with a poset $P$ considered as a topological space with its associated Alexandroff topology. In this paper we show that one can impose such a poset-stratified space structure on the homotopy set $[X, Y]$ of homotopy classes of continuous maps by considering a canonical but non-trivial order (preorder) on it, namely we can capture the homotopy set $[X, Y]$ as an object of the category of poset-stratified spaces. The order we consider is related to the notion of \emph{dependence of maps} (by Karol Borsuk). Furthermore via homology and cohomology the homotopy set $[X,Y]$ can have other poset-stratified space structures. In the cohomology case, we get some results which are equivalent to the notion of \emph{dependence of cohomology classes} (by Ren\'e Thom) and we can show that the set of isomorphism classes of complex vector bundles can be captured as a poset-stratified space via the poset of the subrings consisting of all the characteristic classes. We also show that some invariants such as Gottlieb groups and Lusternik--Schnirelmann category of a map give poset-stratified space structures to the homotopy set $[X,Y]$