

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.
[Gaj96]
Paweł Gajer. The intersection DoldThom
theorem.
Topology
, 35(4):939–967, 1996. Corrigendum: “The
intersection DoldThom theorem” [Topology
35
(1996), no. 4, 939–967; MR 97i:55013].
[GM80]
Mark Goresky and Robert MacPherson. Intersection
homology theory.
Topology
, 19(2):135–162, 1980.
[GM83]
Mark Goresky and Robert MacPherson. Intersection
homology. II.
Invent. Math.
, 72(1):77–129, 1983.
[Tre]
David Treumann. Exit paths and constructible stacks,
arXiv:0708.0659
.
[Wooa]
Jonathan Woolf. The fundamental category of a
stratiﬁed space,
arXiv:0811.2580
.
[Woob]
Jonathan Woolf. Transversal homotopy theory,
arXiv:0910.3322
.


