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Feb, 2020
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Homotopy Theory of Stratified Spaces and Filtered Spaces

The study of manifolds have been the cetral theme in topology, ever since its birth at the end of 19th century. It is a natural idea to try to extend what we know about manifolds to singular spaces.

Goresky and MacPherson [ GM80 GM83 ] introduced intersection homology and cohomology in order to extend the Poincaré duality to singular spaces.

Since then, important tools in algebraic topology such as homotopy groups have been imported to study stratified spaces.

I don’t know, however, the intersection homotopy groups introduced by Gajer [ Gaj96 ] is an appropriate tool.

It might be the right idea to relax the invertibility of paths and define fundamental category instead of fundamental groups or groupoids as is done by Treumann [ Tre ] and Woolf [ Wooa ] . Then how can we extend this idea to define analogues of higher homotopy groups . Woolf’s [ Woob ] defines higher homotopy groups by relaxing invertibility as transversal homotopy monoid.

How can we relate Gajer’s intersection homotopy groups and Woolf’s transversal homotopy monoids.