Topologically stratified space

In topology, a branch of mathematics, a topologically stratified space is a space X that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Topologically stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Whitney. They were introduced by René Thom, who showed that every Whitney stratified space was also a topologically stratified space, with the same strata. Another proof was given by John Mather in 1970, inspired by Thom's proof. Basic examples of stratified spaces include manifolds with boundary (top dimension and codimension 1 boundary) and manifolds with corners (top dimension, codimension 1 boundary, codimension 2 corners).

## Contents

* 1 Definition
* 2 Examples
* 3 See also
* 4 References

## Definition

The definition is inductive on the dimension of X. An n-dimensional topological stratification of X is a filtration

∅=X−1⊂X0⊂X1…⊂Xn=X

of X by closed subspaces such that for each i and for each point x of

Xi∖Xi−1,

there exists a neighborhood

U⊂X

of x in X, a compact (n \- i \- 1)-dimensional stratified space L, and a filtration-preserving homeomorphism

U≅ℝi×CL.

Here CL is the open cone on L.

If X is a topologically stratified space, the i-dimensional stratum of X is the space

Xi∖Xi−1.

Connected components of Xi \ Xi-1 are also frequently called strata.

## Examples

One of the original motivations for stratified spaces were decomposing singular spaces into smooth chunks. For example, given a singular variety

, there is a naturally defined subvariety,

Sing(X)

, which is the singular locus. This may not be a smooth variety, so taking the iterated singularity locus

Sing(Sing(X))

will eventually give a natural stratification. A simple algebreogeometric example is the singular hypersurface

> Spec(ℂ[x,y,z]/(x4+y4+z4))←(0,0,0)Spec(ℂ)

where

Spec(−)

is the prime spectrum.

## See also

* Singularity theory
* Whitney conditions
* Stratifold
* Intersection homology

## References

* Goresky, Mark; MacPherson, Robert Stratified Morse theory, Springer-Verlag, Berlin, 1988.
* Goresky, Mark; MacPherson, Robert Intersection homology II, Invent. Math. 72 (1983), no. 1, 77--129.
* Mather, J. Notes on topological stability, Harvard University, 1970.
* Thom, R. Ensembles et morphismes stratifiés, Bulletin of the American Mathematical Society 75 (1969), pp.240-284.
* Weinberger, Shmuel (1994). The topological classification of stratified spaces. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. ISBN 9780226885667. http://www.press.uchicago.edu/ucp/books/book/chicago/T/bo3625837.html.

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