POSET-STRATIFIED SPACE STRUCTURES OF HOMOTOPY SETS

A poset-stratified space is a pair (S, S ->pi P) of a topological space S and a continuous map pi : S -> 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 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 dependence of cohomology classes (by Rene 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].