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The general name for certain ideas of [[General position|general position]] (cf. also [[Transversal mapping|Transversal mapping]]); a concept in linear algebra, differential and geometric topology.
 
The general name for certain ideas of [[General position|general position]] (cf. also [[Transversal mapping|Transversal mapping]]); a concept in linear algebra, differential and geometric topology.
  
a) Two vector subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940001.png" /> of a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940002.png" /> are transversal to one another if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940004.png" /> generate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940005.png" />, that is, if
+
a) Two vector subspaces $  A, B $
 +
of a finite-dimensional vector space $  C $
 +
are transversal to one another if $  A $
 +
and $  B $
 +
generate $  C $,  
 +
that is, if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940006.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm dim} ( A \cap B) +  \mathop{\rm dim}  C  = \
 +
\mathop{\rm dim}  A +  \mathop{\rm dim}  B.
 +
$$
  
b) In the differentiable situation, two submanifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940007.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940008.png" /> are transversal at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t0940009.png" /> if the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400011.png" /> at this point generate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400012.png" />. Geometrically (for submanifolds in the narrow sense of the word and without boundary) this means that it is possible to introduce local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400013.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400014.png" /> in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400016.png" />, in terms of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400018.png" /> are represented as transversal vector subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400019.png" />.
+
b) In the differentiable situation, two submanifolds $  L, M $
 +
of a manifold $  N $
 +
are transversal at a point $  x \in L \cap M $
 +
if the tangent spaces $  T _ {x} L $,  
 +
$  T _ {x} M $
 +
at this point generate $  T _ {x} N $.  
 +
Geometrically (for submanifolds in the narrow sense of the word and without boundary) this means that it is possible to introduce local coordinates $  x _ {1} \dots x _ {n} $
 +
into $  N $
 +
in some neighbourhood $  U $
 +
of $  x $,  
 +
in terms of which $  L \cap U $
 +
and $  M \cap U $
 +
are represented as transversal vector subspaces of $  \mathbf R  ^ {n} $.
  
A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400020.png" /> is transversal to a submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400021.png" /> at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400022.png" /> (cf. [[Transversal mapping|Transversal mapping]]) if the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400023.png" /> under the differential of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400024.png" /> is transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400026.png" />. Two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400028.png" /> are transversal to each other at a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400030.png" />, if the images of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400032.png" /> generate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400033.png" />. The latter two definitions can also be rephrased geometrically [[#References|[1]]]. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400034.png" /> is transversal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400035.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400036.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400037.png" /> (in old terminology: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400038.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400039.png" />-regular along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400040.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400041.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400042.png" />, if the corresponding transversality holds at all points for which it is possible to talk about it. These concepts easily reduce to one another. E.g. the transversality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400044.png" /> is equivalent to the transversality of the identity imbeddings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400046.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400047.png" />. In common use are the notations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400049.png" />, etc.
+
A mapping $  f: L \rightarrow N $
 +
is transversal to a submanifold $  M \subset  N $
 +
at a point $  x \in f ^ { - 1 } ( M) $(
 +
cf. [[Transversal mapping|Transversal mapping]]) if the image of $  T _ {x} L $
 +
under the differential of $  f $
 +
is transversal to $  T _ {f ( x) }  M $
 +
in $  T _ {f ( x) }  N $.  
 +
Two mappings $  f: L \rightarrow N $
 +
and $  g: M \rightarrow N $
 +
are transversal to each other at a point $  ( x, y) \in L \times M $,  
 +
where $  f ( x) = g ( y) $,  
 +
if the images of $  T _ {x} L $
 +
and $  T _ {y} M $
 +
generate $  T _ {f ( x) }  N $.  
 +
The latter two definitions can also be rephrased geometrically [[#References|[1]]]. One says that $  L $
 +
is transversal to $  M $,  
 +
and $  f $
 +
to $  M $(
 +
in old terminology: $  f $
 +
is t $-
 +
regular along $  M $),  
 +
and $  f $
 +
to $  g $,  
 +
if the corresponding transversality holds at all points for which it is possible to talk about it. These concepts easily reduce to one another. E.g. the transversality of $  L $
 +
and $  M $
 +
is equivalent to the transversality of the identity imbeddings of $  L $
 +
and $  M $
 +
in $  N $.  
 +
In common use are the notations $  L \cap _ {x} M $,  
 +
$  f \cap M $,  
 +
etc.
  
 
For transversality of manifolds with boundary it is sometimes useful to require certain extra conditions to hold (see [[#References|[3]]]). Transversality also carries over to the infinite-dimensional case (see [[#References|[1]]], [[#References|[2]]]).
 
For transversality of manifolds with boundary it is sometimes useful to require certain extra conditions to hold (see [[#References|[3]]]). Transversality also carries over to the infinite-dimensional case (see [[#References|[1]]], [[#References|[2]]]).
  
In all these situations the role of transversality is connected with  "genericity"  and with the  "good"  properties of the intersection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400050.png" />, the pre-images <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094000/t09400051.png" />, and analogous objects (which are deformed to the same  "good"  objects, if under the deformation of the original objects transversality is preserved) (see [[#References|[4]]]).
+
In all these situations the role of transversality is connected with  "genericity"  and with the  "good"  properties of the intersection $  L \cap M $,  
 +
the pre-images $  f ^ { - 1 } ( M) $,  
 +
and analogous objects (which are deformed to the same  "good"  objects, if under the deformation of the original objects transversality is preserved) (see [[#References|[4]]]).
  
 
c) In piecewise-linear and topological situations, transversality of submanifolds is defined similarly to the geometric definition in b). (Especially widespread is the piecewise-linear version for submanifolds of complementary dimension, cf. [[#References|[5]]].) In general, one does not obtain a complete analogy with the properties of transversality in b) (see [[#References|[6]]], [[#References|[8]]]), therefore more restricted modifications of transversality have been proposed (see [[#References|[7]]], [[#References|[9]]]).
 
c) In piecewise-linear and topological situations, transversality of submanifolds is defined similarly to the geometric definition in b). (Especially widespread is the piecewise-linear version for submanifolds of complementary dimension, cf. [[#References|[5]]].) In general, one does not obtain a complete analogy with the properties of transversality in b) (see [[#References|[6]]], [[#References|[8]]]), therefore more restricted modifications of transversality have been proposed (see [[#References|[7]]], [[#References|[9]]]).

Latest revision as of 08:26, 6 June 2020


The general name for certain ideas of general position (cf. also Transversal mapping); a concept in linear algebra, differential and geometric topology.

a) Two vector subspaces $ A, B $ of a finite-dimensional vector space $ C $ are transversal to one another if $ A $ and $ B $ generate $ C $, that is, if

$$ \mathop{\rm dim} ( A \cap B) + \mathop{\rm dim} C = \ \mathop{\rm dim} A + \mathop{\rm dim} B. $$

b) In the differentiable situation, two submanifolds $ L, M $ of a manifold $ N $ are transversal at a point $ x \in L \cap M $ if the tangent spaces $ T _ {x} L $, $ T _ {x} M $ at this point generate $ T _ {x} N $. Geometrically (for submanifolds in the narrow sense of the word and without boundary) this means that it is possible to introduce local coordinates $ x _ {1} \dots x _ {n} $ into $ N $ in some neighbourhood $ U $ of $ x $, in terms of which $ L \cap U $ and $ M \cap U $ are represented as transversal vector subspaces of $ \mathbf R ^ {n} $.

A mapping $ f: L \rightarrow N $ is transversal to a submanifold $ M \subset N $ at a point $ x \in f ^ { - 1 } ( M) $( cf. Transversal mapping) if the image of $ T _ {x} L $ under the differential of $ f $ is transversal to $ T _ {f ( x) } M $ in $ T _ {f ( x) } N $. Two mappings $ f: L \rightarrow N $ and $ g: M \rightarrow N $ are transversal to each other at a point $ ( x, y) \in L \times M $, where $ f ( x) = g ( y) $, if the images of $ T _ {x} L $ and $ T _ {y} M $ generate $ T _ {f ( x) } N $. The latter two definitions can also be rephrased geometrically [1]. One says that $ L $ is transversal to $ M $, and $ f $ to $ M $( in old terminology: $ f $ is $ t $- regular along $ M $), and $ f $ to $ g $, if the corresponding transversality holds at all points for which it is possible to talk about it. These concepts easily reduce to one another. E.g. the transversality of $ L $ and $ M $ is equivalent to the transversality of the identity imbeddings of $ L $ and $ M $ in $ N $. In common use are the notations $ L \cap _ {x} M $, $ f \cap M $, etc.

For transversality of manifolds with boundary it is sometimes useful to require certain extra conditions to hold (see [3]). Transversality also carries over to the infinite-dimensional case (see [1], [2]).

In all these situations the role of transversality is connected with "genericity" and with the "good" properties of the intersection $ L \cap M $, the pre-images $ f ^ { - 1 } ( M) $, and analogous objects (which are deformed to the same "good" objects, if under the deformation of the original objects transversality is preserved) (see [4]).

c) In piecewise-linear and topological situations, transversality of submanifolds is defined similarly to the geometric definition in b). (Especially widespread is the piecewise-linear version for submanifolds of complementary dimension, cf. [5].) In general, one does not obtain a complete analogy with the properties of transversality in b) (see [6], [8]), therefore more restricted modifications of transversality have been proposed (see [7], [9]).

Finally, a category of manifolds is said to have the transversality property if any mapping in it can be approximated by a transversal mapping.

References

[1] S. Lang, "Introduction to differentiable manifolds" , Interscience (1967)
[2] N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French)
[3] V.A. Rokhlin, D.B. Fuks, "Beginner's course in topology. Geometric chapters" , Springer (1984) (Translated from Russian)
[4] M.W. Hirsch, "Differential topology" , Springer (1976)
[5] C.P. Rourke, B.J. Sanderson, "Introduction to piecewise-linear topology" , Springer (1972)
[6] W. Lickorish, C.P. Rourke, "A counter-example to the three balls problem" Proc. Cambridge Philos. Soc. , 66 (1969) pp. 13–16
[7] C.P. Rourke, B.J. Sanderson, "Block bundles II. Transversality" Ann. of Math. , 87 (1968) pp. 256–278
[8] J.F.P. Hudson, "On transversality" Proc. Cambridge Philos. Soc. , 66 (1969) pp. 17–20
[9] A. Marin, "La transversalité topologique" Ann. of Math. , 106 : 2 (1977) pp. 269–293
How to Cite This Entry:
Transversality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transversality&oldid=49026
This article was adapted from an original article by D.V. Anosov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article