Stratification

A decomposition of a (possibly infinite-dimensional) manifold into connected submanifolds of strictly-diminishing dimensions.

Comments

Usually a "stratification" of a space means more than just some decomposition into connected pieces with diminishing dimensions.

Let $ ( P, < ) $ be a partially ordered set. A $ P $- decomposition of a topological space $ X $ is a locally finite collection of subspaces $ S _ {i} $ of $ X $, labelled by the elements of $ P $, such that:

1) $ S _ {i} \cap S _ {j} = \emptyset $ if $ i \neq j $;

2) $ S _ {i} $ is locally closed for all $ i \in P $;

3) $ X = \cup _ {i \in P } S _ {i} $;

4) if $ S _ {i} \cap \overline{ {S _ {j} }}\; \neq \emptyset $, then $ S _ {i} \subset \overline{ {S _ {j} }}\; $( and this is equivalent to $ i \leq j $ in $ P $).

As an example, consider the subset of $ \mathbf R ^ {2} $ given by the inequality $ x ^ {3} - y ^ {2} \geq 0 $ divided into the four pieces $ \{ {( x, y) } : {x ^ {3} - y ^ {2} > 0 } \} $, $ \{ {( x, y) } : {x ^ {3} = y ^ {2} , y > 0 } \} $, $ \{ {( x, y) } : {x ^ {3} = y ^ {2} , y < 0 } \} $, $ \{ ( 0, 0) \} $.

Now let $ X $ be a subset of a smooth manifold $ M $. A stratification of $ X $ is a $ P $- decomposition $ ( S _ {i} ) _ {i \in P } $ for some partially ordered set $ P $ such that each of the pieces is a smooth submanifold of $ M $.

The stratification $ ( S _ {i} ) $ is called a Whitney stratification if for every pair of strata $ S _ {i} , S _ {j} $ with $ S _ {i} \subset \overline{ {S _ {j} }}\; $ the following Whitney's conditions $ A $ and $ B $ hold. Suppose that a sequence of points $ y _ {k} \in S _ {i} $ converges to $ y \in S _ {i} $ and a sequence of points $ x _ {k} \in S _ {j} $ also converges to $ y \in S _ {i} $. Suppose, moreover, that the tangent planes $ T _ {x _ {k} } S _ {j} $ converge to some limiting plane $ T $ and that the secant lines $ \overline{ {x _ {k} y _ {k} }}\; $ converge to some line $ l $( all with respect to some local coordinate system around $ y $ in the ambient manifold $ M $). Then

A) $ T _ {y} S _ {i} \subset T $;

B) $ l \subset T $.

Condition B) implies in fact condition A).

A few facts and theorems concerning Whitney stratifications are as follows. Any closed subanalytic subset of an analytic manifold admits a Whitney stratification, [a5]. In particular, algebraic sets in $ \mathbf R ^ {n} $, i.e. sets given by the vanishing of finitely many polynomials (cf. also Semi-algebraic set), can be Whitney stratified. Whitney stratified spaces can be triangulated, [a4].

References

[a1]	J. Mather, "Notes on topological stability" , Harvard Univ. Press (1970) (Mimeographed notes)
[a2]	C.G. Gibson, K. Wirthmüller, A.A. du Plessis, E.J.N. Looijenga, "Topological stability of smooth mappings" , Lect. notes in math. , 552 , Springer (1976) MR0436203 Zbl 0377.58006
[a3]	M. Goresky, "Stratified Morse theory" , Springer (1988) MR0932724 Zbl 0639.14012
[a4]	F. Johnson, "On the triangulation of stratified sets and singular varieties" Trans. Amer. Math. Soc. , 275 (1983) pp. 333–343 MR0678354 Zbl 0511.58007
[a5]	H. Hironaka, "Subanalytic sets" , Number theory, algebraic geometry and commutative algebra , Kinokuniya (1973) pp. 453–493 MR0377101 Zbl 0297.32008
[a6]	H. Whitney, "Tangents to an analytic variety" Ann. of Math. , 81 (1965) pp. 496–549 MR0192520 Zbl 0152.27701
[a7]	H. Whitney, "Local properties of analytic varieties" S. Cairns (ed.) , Differentiable and Combinatorial Topology , Princeton Univ. Press (1965) pp. 205–244 MR0188486 Zbl 0129.39402
[a8]	R. Thom, "Propriétés différentielles locales des ensembles analytiques" , Sem. Bourbaki , Exp. 281 (1964/5) MR1608789 Zbl 0184.31402