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Sylvain Douteau: Stratified homotopy theory

Time: Wed 2021-10-27 10.15 - 11.15

Location: Kräftriket, House 6, Room 306

Lecturer: Sylvain Douteau (Stockholm University)

Abstract: Whitney's theorem (1965), stating that any singular complex algebraic variety can be decomposed into smooth manifolds, satisfying gluing conditions, is at the origin of the study of stratified spaces. Since then, numerous invariants of manifolds have been extended to pseudo-manifolds — those objects that Whitney defined — among them, intersection cohomology, introduced by Goresky and MacPherson in 1980. Invariants of pseudo-manifolds tend to only be invariants up to stratum preserving homotopies. This observation motivated Goresky and MacPherson to ask what was the corresponding homotopy theory.

A homotopy theory for stratified spaces should satisfy several conditions. First, it should be defined on a large enough category to allow for the usual constructions in homotopy theory — in particular, it has to contain objects that are not pseudo-manifolds. On the other hand, it is known that the stratified homotopy type of a pseudo-manifold can be recovered from the data of the strata and of gluing “instructions”. The same should be true for any stratified space. Finally, the homotopy theory of pseudo-manifolds should be compatible with that of all stratified spaces.

In this talk, I will explain how one can get such a stratified homotopy theory by working with suitable generalizations of the homotopy links, defined by Quinn in 1988 for homotopically stratified sets.