In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, 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 f : X → Y {\displaystyle f\colon X\rightarrow Y} , may be deformed by an arbitrarily small amount into a map that is transverse to a given submanifold Z ⊆ Y {\displaystyle Z\subseteq Y} . 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.
Finite-dimensional version
Previous definitions Let f : X → Y {\displaystyle f\colon X\rightarrow Y} be a smooth map between smooth manifolds, and let Z {\displaystyle Z} be a submanifold of Y {\displaystyle Y} . We say that f {\displaystyle f} is transverse to Z {\displaystyle Z} , denoted as f ⋔ Z {\displaystyle f\pitchfork Z} , if and only if for every x ∈ f − 1 ( Z ) {\displaystyle x\in f^{-1}\left(Z\right)} we have that
im ( d f x ) + T f ( x ) Z = T f ( x ) Y {\displaystyle \operatorname {im} \left(df_{x}\right)+T_{f\left(x\right)}Z=T_{f\left(x\right)}Y} . An important result about transversality states that if a smooth map f {\displaystyle f} is transverse to Z {\displaystyle Z} , then f − 1 ( Z ) {\displaystyle f^{-1}\left(Z\right)} is a regular submanifold of X {\displaystyle X} . If X {\displaystyle X} is a manifold with boundary, then we can define the restriction of the map f {\displaystyle f} to the boundary, as ∂ f : ∂ X → Y {\displaystyle \partial f\colon \partial X\rightarrow Y} . The map ∂ f {\displaystyle \partial f} is smooth, and it allows us to state an extension of the previous result: if both f ⋔ Z {\displaystyle f\pitchfork Z} and ∂ f ⋔ Z {\displaystyle \partial f\pitchfork Z} , then f − 1 ( Z ) {\displaystyle f^{-1}\left(Z\right)} is a regular submanifold of X {\displaystyle X} with boundary, and
∂ f − 1 ( Z ) = f − 1 ( Z ) ∩ ∂ X {\displaystyle \partial f^{-1}\left(Z\right)=f^{-1}\left(Z\right)\cap \partial X} .
… excerpt ends here. Continue reading the full article.
