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''transversally-regular mapping''

A mapping with certain properties of [[General position|general position]].

Let  $  \xi $
be a [[Vector bundle|vector bundle]] over a finite [[Cellular space|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|normal bundle]]  $  \nu $
and if the restriction of  $  f $
to a [[Tubular neighbourhood|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|Transversality]]).

The approximation theorem [[#References|[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|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 [[#References|[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 [[#References|[4]]] and for  $   \mathop{\rm dim}  M = 4 $
in [[#References|[5]]], based on a subsequently proved, difficult result of [[#References|[6]]]. The notion of a transversal mapping can also be formulated for infinite-dimensional manifolds.

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

====Comments====
The notion of [[Transversality|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 [[#References|[a1]]]–[[#References|[a3]]].

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