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).

Definition

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

[math]\displaystyle{ \emptyset = X_{-1} \subset X_0 \subset X_1 \ldots \subset X_n = X }[/math]

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

[math]\displaystyle{ X_i \setminus X_{i-1} }[/math],

there exists a neighborhood

[math]\displaystyle{ U \subset X }[/math]

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

[math]\displaystyle{ U \cong \mathbb{R}^i \times CL }[/math].

Here [math]\displaystyle{ CL }[/math] is the open cone on L.

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

[math]\displaystyle{ X_i \setminus X_{i-1} }[/math].

Connected components of X_i \ X_i-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 [math]\displaystyle{ X }[/math], there is a naturally defined subvariety, [math]\displaystyle{ Sing(X) }[/math], which is the singular locus. This may not be a smooth variety, so taking the iterated singularity locus [math]\displaystyle{ Sing(Sing(X)) }[/math]will eventually give a natural stratification. A simple algebreogeometric example is the singular hypersurface

[math]\displaystyle{ \text{Spec}(\mathbb{C}[x,y,z]/(x^4 + y^4 + z^4)) \xleftarrow{(0,0,0)} \text{Spec}(\mathbb{C}) }[/math]

where [math]\displaystyle{ \text{Spec}(-) }[/math] 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.