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 topological 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.
The definition is inductive on the dimension of X. An n-dimensional topological stratification of X is a filtration
of X by closed subspaces such that for each i and for each point x of
there exists a neighborhood
of x in X, a compact (n - i - 1)-dimensional stratified space L, and a filtration-preserving homeomorphism
Here is the open cone on L.
If X is a topologically stratified space, the i-dimensional stratum of X is the space
Connected components of Xi \ Xi-1 are also frequently called strata.
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, , which is the singular locus. This may not be a smooth variety, so taking the iterated singularity locus will eventually give a natural stratification. A simple algebreogeometric example is the singular hypersurface
where is the prime spectrum.
- 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.
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