Topologically stratified space

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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 Mark Goresky and Robert MacPherson.

[edit] Definition

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

X_i \ X_i-1,

there exists a neighborhood

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

.

Here CL is the open cone on L.

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

X_i \ X_i-1.

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

[edit] See also

• Singularity theory
• Whitney conditions
• Thom-Mather stratified space
• Intersection homology

[edit] References

• Goresky, Mark; MacPherson, Robert Intersection homology II. Invent. Math. 72 (1983), no. 1, 77--129.