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The general name for certain ideas of [[General position|general position]] (cf. also [[Transversal mapping|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|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 [[#References|[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 [[#References|[3]]]). Transversality also carries over to the infinite-dimensional case (see [[#References|[1]]], [[#References|[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 [[#References|[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. [[#References|[5]]].) In general, one does not obtain a complete analogy with the properties of transversality in b) (see [[#References|[6]]], [[#References|[8]]]), therefore more restricted modifications of transversality have been proposed (see [[#References|[7]]], [[#References|[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|transversal mapping]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lang,   "Introduction to differentiable manifolds" , Interscience  (1967)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley  (1966)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Rokhlin,   D.B. Fuks,   "Beginner's course in topology. Geometric chapters" , Springer  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M.W. Hirsch,   "Differential topology" , Springer  (1976)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  C.P. Rourke,   B.J. Sanderson,   "Introduction to piecewise-linear topology" , Springer  (1972)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  W. Lickorish,   C.P. Rourke,   "A counter-example to the three balls problem"  ''Proc. Cambridge Philos. Soc.'' , '''66'''  (1969)  pp. 13–16</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  C.P. Rourke,   B.J. Sanderson,   "Block bundles II. Transversality"  ''Ann. of Math.'' , '''87'''  (1968)  pp. 256–278</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J.F.P. Hudson,   "On transversality"  ''Proc. Cambridge Philos. Soc.'' , '''66'''  (1969)  pp. 17–20</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Marin,   "La transversalité topologique"  ''Ann. of Math.'' , '''106''' :  2  (1977)  pp. 269–293</TD></TR></table>