# Stratification

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

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
How to Cite This Entry:
Stratification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stratification&oldid=48868
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article