Transversal mapping

transversally-regular mapping

A mapping with certain properties of general position.

Let $ \xi $ be a vector bundle over a finite cellular space $ X $, and let the total space of $ \xi $ be imbedded as an open subset in some topological space $ Z $. Then a continuous mapping $ f: M \rightarrow Z $, where $ M $ is a smooth manifold, is called a transversal mapping to $ X $ if $ V = f ^ { - 1 } ( X) $ is a smooth submanifold of $ M $ with normal bundle $ \nu $ and if the restriction of $ f $ to a tubular neighbourhood of $ V $ in $ M $ defines a morphism of bundles $ \nu \rightarrow \xi $.

For example, let $ f: M \rightarrow N $ be a smooth mapping of smooth manifolds, and let $ X $ be a smooth submanifold of $ N $. If the differential $ df: \tau _ {M} \rightarrow \tau _ {N} $( where $ \tau $ is the tangent bundle) contains in its image all vectors normal to $ X $ in $ N $ of the bundle $ \xi $, then $ f $ is a transversal mapping (cf. also Transversality).

The approximation theorem [1]: The transversal mappings form a set of the second category in the set of all continuous mappings $ M \rightarrow Z $. Moreover, any continuous mapping is homotopic to a transversal mapping. This theorem enables one to associate with algebraic invariants (homotopy classes of mappings) descriptive geometric forms (certain equivalence classes of manifolds that are pre-images under transversal mappings). This association also goes in the other direction, namely from geometry to algebra. Along these lines, various bordism groups, for example, have been calculated, smooth manifolds of given homotopy type have been classified, etc.

The notion of a transversal mapping can be carried over to the categories of piecewise-linear and topological manifolds and $ \mathbf R ^ {n} $- bundles. Furthermore, in the piecewise-linear category the approximation theorem holds; see [3]. Also, in the topological category every continuous mapping is homotopic to a transversal one; this was proved for $ \mathop{\rm dim} M \neq 4 \neq \mathop{\rm dim} M - \mathop{\rm dim} \xi $ in [4] and for $ \mathop{\rm dim} M = 4 $ in [5], based on a subsequently proved, difficult result of [6]. The notion of a transversal mapping can also be formulated for infinite-dimensional manifolds.

References

[1]	R. Thom, "Un lemma sur les applications différentiables" Bol. Soc. Mat. Mex. , 1 (1956) pp. 59–71
[2]	W.B. Browder, "Surgery on simply connected manifolds" , Springer (1972)
[3]	R. Williamson, "Cobordism of combinatorial manifolds" Ann. of Math. , 83 (1966) pp. 1–33
[4]	R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothing, and triangulations" , Princeton Univ. Press (1977)
[5]	M. Sharlemann, "Transversality theories at dimension 4" Invent. Math. , 33 (1976) pp. 1–14
[6]	M. Freedman, "The topology of four-dimensional manifolds" J. Diff. Geom. , 17 (1982) pp. 357–453

Comments

The notion of transversality is defined for arbitrary smooth mappings $ f : M \rightarrow N $ between smooth manifolds. If $ A $ is a smooth submanifold of $ N $, then $ f $ is transverse to $ A $ if for every $ x $ in the pre-image $ V $ of $ A $, the tangent space to $ N $ at $ f ( x) $ is spanned by the tangent space to $ A $ at $ f ( x) $ and the image, under the differential of $ f $, of the tangent space to $ M $ at $ x $. When this holds, then $ V $ is a smooth submanifold of $ M $, and the normal bundle to $ V $ in $ M $ is the pull-back under $ f $ of the normal bundle to $ A $ in $ N $. The approximation theorem is valid for such mappings. For the use of transversality in topology see [a1]–[a3].

References

[a1]	L.S. Pontryagin, "Smooth manifolds and their applications in homotopy theory" , Amer. Math. Soc. (1959) (Translated from Russian)
[a2]	R. Thom, "Quelques propriétés globales des variétés différentiables" Comment. Math. Helvetica , 28 (1954) pp. 17–86
[a3]	M.W. Hirsch, "Differential topology" , Springer (1976) pp. 4, 78