Stratified path spaces and fibrations

Abstract

Abstract. The main objects of study are the homotopically stratified metric spaces introduced by Quinn. Closed unions of strata are shown to be stratified forward tame. Stratified fibrations between spaces with stratifications are introduced. Paths which lie in a single stratum except possibly at their initial points form a space with a natural stratification, and the evaluation map from that space of paths is shown to be a stratified fibration. Applications to mapping cylinders and to the geometry of manifold stratified spaces are expected in future papers.

Intersection homology and Poincaré duality on homotopically stratified spaces

“…If the skeleta X i are closed in X , then these spaces will also be "stratified spaces satisfying the frontier condition" -see Hughes [27].…”

“…p 1 .U / and F W Z I ! U for some metric space Z (we may assume Z to be metric by Remark 5.5b of [27]). Certainly there is a weak stratified controlled solution z F W Z I OE0; 1/ !…”

“…which is a stratified fibration by [27], and let P be the mapping cylinder collapse, which is also a stratified fibration by [17,Proposition 3.3]. We will show below that IH c .…”

“…U \ X n k is a stratified fibration by [27,Corollary 6.2] and [17,Proposition 3.3]. Furthermore, since U \ X n k is homeomorphic to R n k , by [16,Corollary 3.14], M U is stratum-and fiber-preserving homotopy equivalent to R n k times the fiber over x , which is the cone on holink.X; x/.…”

Hochschild homology relative to a family of groups

“…We recall some of the basic definitions from the theory of stratified spaces following the treatment in Hughes [11].…”