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A decomposition of a (possibly infinite-dimensional) manifold into connected submanifolds of strictly-diminishing dimensions.
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A decomposition of a (possibly infinite-dimensional) manifold into connected submanifolds of strictly-diminishing dimensions.
  
 
====Comments====
 
====Comments====
 
Usually a "stratification" of a space means more than just some decomposition into connected pieces with diminishing dimensions.
 
Usually a "stratification" of a space means more than just some decomposition into connected pieces with diminishing dimensions.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s0904301.png" /> be a partially ordered set. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s0904303.png" />-decomposition of a topological space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s0904304.png" /> is a locally finite collection of subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s0904305.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s0904306.png" />, labelled by the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s0904307.png" />, such that:
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Let $  ( P, < ) $
 +
be a partially ordered set. A $  P $-
 +
decomposition of a topological space $  X $
 +
is a locally finite collection of subspaces $  S _ {i} $
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of $  X $,  
 +
labelled by the elements of $  P $,  
 +
such that:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s0904308.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s0904309.png" />;
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1) $  S _ {i} \cap S _ {j} = \emptyset $
 +
if $  i \neq j $;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043010.png" /> is locally closed for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043011.png" />;
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2) $  S _ {i} $
 +
is locally closed for all $  i \in P $;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043012.png" />;
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3) $  X = \cup _ {i \in P }  S _ {i} $;
  
4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043013.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043014.png" /> (and this is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043015.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043016.png" />).
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043017.png" /> given by the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043018.png" /> divided into the four pieces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043022.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043023.png" /> be a subset of a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043024.png" />. A stratification of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043025.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043026.png" />-decomposition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043027.png" /> for some partially ordered set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043028.png" /> such that each of the pieces is a smooth submanifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043029.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043030.png" /> is called a Whitney stratification if for every pair of strata <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043031.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043032.png" /> the following Whitney's conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043036.png" /> hold. Suppose that a sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043037.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043038.png" /> and a sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043039.png" /> also converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043040.png" />. Suppose, moreover, that the tangent planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043041.png" /> converge to some limiting plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043042.png" /> and that the secant lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043043.png" /> converge to some line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043044.png" /> (all with respect to some local coordinate system around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043045.png" /> in the ambient manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043046.png" />). Then
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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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043047.png" />;
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A) $  T _ {y} S _ {i} \subset  T $;
  
B) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043048.png" />.
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B) $  l \subset  T $.
  
 
Condition B) implies in fact condition A).
 
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, [[#References|[a5]]]. In particular, algebraic sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090430/s09043049.png" />, i.e. sets given by the vanishing of finitely many polynomials (cf. also [[Semi-algebraic set|Semi-algebraic set]]), can be Whitney stratified. Whitney stratified spaces can be triangulated, [[#References|[a4]]].
+
A few facts and theorems concerning Whitney stratifications are as follows. Any closed subanalytic subset of an analytic manifold admits a Whitney stratification, [[#References|[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|Semi-algebraic set]]), can be Whitney stratified. Whitney stratified spaces can be triangulated, [[#References|[a4]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Mather, "Notes on topological stability" , Harvard Univ. Press (1970) (Mimeographed notes)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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) {{MR|0436203}} {{ZBL|0377.58006}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Goresky, "Stratified Morse theory" , Springer (1988) {{MR|0932724}} {{ZBL|0639.14012}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Johnson, "On the triangulation of stratified sets and singular varieties" ''Trans. Amer. Math. Soc.'' , '''275''' (1983) pp. 333–343 {{MR|0678354}} {{ZBL|0511.58007}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H. Hironaka, "Subanalytic sets" , ''Number theory, algebraic geometry and commutative algebra'' , Kinokuniya (1973) pp. 453–493 {{MR|0377101}} {{ZBL|0297.32008}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> H. Whitney, "Tangents to an analytic variety" ''Ann. of Math.'' , '''81''' (1965) pp. 496–549 {{MR|0192520}} {{ZBL|0152.27701}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> H. Whitney, "Local properties of analytic varieties" S. Cairns (ed.) , ''Differentiable and Combinatorial Topology'' , Princeton Univ. Press (1965) pp. 205–244 {{MR|0188486}} {{ZBL|0129.39402}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Thom, "Propriétés différentielles locales des ensembles analytiques" , ''Sem. Bourbaki'' , '''Exp. 281''' (1964/5) {{MR|1608789}} {{ZBL|0184.31402}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J. Mather, "Notes on topological stability" , Harvard Univ. Press (1970) (Mimeographed notes)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> 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) {{MR|0436203}} {{ZBL|0377.58006}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> M. Goresky, "Stratified Morse theory" , Springer (1988) {{MR|0932724}} {{ZBL|0639.14012}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Johnson, "On the triangulation of stratified sets and singular varieties" ''Trans. Amer. Math. Soc.'' , '''275''' (1983) pp. 333–343 {{MR|0678354}} {{ZBL|0511.58007}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> H. Hironaka, "Subanalytic sets" , ''Number theory, algebraic geometry and commutative algebra'' , Kinokuniya (1973) pp. 453–493 {{MR|0377101}} {{ZBL|0297.32008}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> H. Whitney, "Tangents to an analytic variety" ''Ann. of Math.'' , '''81''' (1965) pp. 496–549 {{MR|0192520}} {{ZBL|0152.27701}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> H. Whitney, "Local properties of analytic varieties" S. Cairns (ed.) , ''Differentiable and Combinatorial Topology'' , Princeton Univ. Press (1965) pp. 205–244 {{MR|0188486}} {{ZBL|0129.39402}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R. Thom, "Propriétés différentielles locales des ensembles analytiques" , ''Sem. Bourbaki'' , '''Exp. 281''' (1964/5) {{MR|1608789}} {{ZBL|0184.31402}} </TD></TR></table>

Latest revision as of 08:23, 6 June 2020


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
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