In
differential topology, the
transversality theorem, also known as the
Thom Transversality Theorem, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It says that
transversality is a
generic property: any smooth map
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=1243)
, may be deformed by an arbitrary small amount into a map that is transverse to a given submanifold
![](http://info.babylon.com/onlinebox.cgi?rt=GetFile&uri=!!ARV6FUJ2JP&type=0&index=1204)
. Together with the
Pontryagin-Thom construction, it is the technical heart of
cobordism theory, and the starting point for
surgery theory. The finite-dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite-dimensional parametrization using the infinite-dimensional version of the transversality theorem.