# 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 : 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 . 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  and for $\mathop{\rm dim} M = 4$ in , based on a subsequently proved, difficult result of . The notion of a transversal mapping can also be formulated for infinite-dimensional manifolds.

How to Cite This Entry:
Transversal mapping. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Transversal_mapping&oldid=49024
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article