# Transversality

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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