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- W3100508419 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 $[X,Y]$" @default.
- W3100508419 created "2020-11-23" @default.
- W3100508419 creator A5062349983 @default.
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- W3100508419 date "2019-01-01" @default.
- W3100508419 modified "2023-09-28" @default.
- W3100508419 title "Poset-stratified space structures of homotopy sets" @default.
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- W3100508419 doi "https://doi.org/10.4310/hha.2019.v21.n2.a1" @default.
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