Poset-stratified space structures of homotopy sets
Toshihiro Yamaguchi, Shoji Yokura

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 of a topological space and a continuous map with a poset 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 of homotopy classes of continuous maps by considering a canonical but non-trivial order (preorder) on it, namely we can capture the homotopy set 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 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…
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