Transversality

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 of a finite-dimensional vector space are transversal to one another if and generate , that is, if

b) In the differentiable situation, two submanifolds of a manifold are transversal at a point if the tangent spaces , at this point generate . Geometrically (for submanifolds in the narrow sense of the word and without boundary) this means that it is possible to introduce local coordinates into in some neighbourhood of , in terms of which and are represented as transversal vector subspaces of .

A mapping is transversal to a submanifold at a point (cf. Transversal mapping) if the image of under the differential of is transversal to in . Two mappings and are transversal to each other at a point , where , if the images of and generate . The latter two definitions can also be rephrased geometrically [1]. One says that is transversal to , and to (in old terminology: is -regular along ), and to , 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 and is equivalent to the transversality of the identity imbeddings of and in . In common use are the notations , , 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 , the pre-images , 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