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 $ 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