Stratified space

Short description: Type of topological space

In mathematics, especially in topology, a stratified space is a topological space that admits or is equipped with a stratification, a decomposition into subspaces, which are nice in some sense (e.g., smooth or flat).

A basic example is a subset of a smooth manifold that admits a Whitney stratification. But there is also an abstract stratified space such as a Thom–Mather stratified space.

On a stratified space, a constructible sheaf can be defined as a sheaf that is locally constant on each stratum.

Among the several ideals, Grothendieck's Esquisse d’un programme considers (or proposes) a stratified space with what he calls the tame topology.

A stratified space in the sense of Mather

Mather gives the following definition of a stratified space. A prestratification on a topological space X is a partition of X into subsets (called strata) such that (a) each stratum is locally closed, (b) it is locally finite and (c) (axiom of frontier) if two strata A, B are such that the closure of A intersects B, then B lies in the closure of A. A stratification on X is a rule that assigns to a point x in X a set germ [math]\displaystyle{ S_x }[/math] at x of a closed subset of X that satisfies the following axiom: for each point x in X, there exists a neighborhood U of x and a prestratification of U such that for each y in U, [math]\displaystyle{ S_x }[/math] is the set germ at y of the stratum of the prestratification on U containing y.

A stratified space is then a topological space equipped with a stratification.

Pseudomanifold

In the MacPherson's stratified pseudomanifolds; the strata are the differences X_i+i-X_i between sets in the filtration. There is also a local conical condition; there must be an almost smooth atlas where locally each little open set looks like the product of two factors Rⁿx c(L); a euclidean factor and the topological cone of a space L. Classically, here is the point where the definitions turns to be obscure, since L is asked to be a stratified pseudomanifold. The logical problem is avoided by an inductive trick which makes different the objects L and X.

The changes of charts or cocycles have no conditions in the MacPherson's original context. Pflaum asks them to be smooth, while in the Thom-Mather context they must preserve the above decomposition, they have to be smooth in the Euclidean factor and preserve the conical radium.

See also

• Equisingularity
• Perverse sheaf
• Stratified Morse theory
• Harder–Narasimhan stratification

References

• Appendix 1 of R. MacPherson, Intersection homology and perverse sheaves, 1990 notes
• J. Mather, Stratifications and Mappings, Dynamical Systems, Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26–August 14, 1971, 1973, pages 195–232.
• Markus J. Pflaum, Analytic and Geometric Study of Stratified Spaces: Contributions to Analytic and Geometric Aspects (Lecture Notes in Mathematics, 1768) ; Publisher, Springer;

Further reading

• https://ncatlab.org/nlab/show/stratified+space
• https://mathoverflow.net/questions/258562/correct-definition-of-stratified-spaces-and-reference-for-constructible-sheave
• Chapter 2 of Greg Friedman, Singular intersection homology
• https://ncatlab.org/nlab/show/poset-stratified+space