Strongly stratified homotopy theory (Q2847128)

The present paper defines the notion of a \textit{strongly stratified map} between filtered spaces and then investigates homotopy theoretic consequences for homotopically stratified spaces in the sense of Frank Quinn. A map is strongly stratified if domain and target spaces have the same number of strata, the preimage of any stratum of the target is a stratum of the domain, and the map induces an isomorphism of partial orders determined by the filtrations. The main result is that one can decide for such a map between homotopically stratified spaces whether it is a strongly stratified homotopy equivalence or not by looking at strata individually and by looking at the homotopy-links (``holinks'') between the strata. If the restrictions of the map to all strata are strongly stratified homotopy equivalences and the induced maps on holinks are all strongly stratified homotopy equivalences, then so is the map itself.