Stratification

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

Comments[edit]

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) $S_i\cap S_j=\mathrm{\varnothing }$ if $i\ne j$;

2) $S_i$ is locally closed for all $i\in P$;

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

4) if $S_i\cap \overline{S_j}\ne \mathrm{\varnothing }$, then $S_i\subset \overline{S_j}$( and this is equivalent to $i\le j$ in $P$).

As an example, consider the subset of ${\mathbf{R}}^2$ given by the inequality $x^3-y^2\ge 0$ 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].

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