Thom's first isotopy lemma
In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map between smooth manifolds and a closed Whitney stratified subset, if is proper and is a submersion for each stratum of, then is a locally trivial fibration. The lemma was originally introduced by René Thom who considered the case when. In that case, the lemma constructs an isotopy from the fiber to ; whence the name "isotopy lemma".
The local trivializations that the lemma provide preserve the strata. However, they are generally not smooth. On the other hand, it is possible that local trivializations are semialgebraic if the input data is semialgebraic.
The lemma is also valid for a more general stratified space such as a stratified space in the sense of Mather but still with the Whitney conditions. The lemma is also valid for the stratification that satisfies Bekka's condition (C), which is weaker than Whitney's condition.
Thom's second isotopy lemma is a family version of the first isotopy lemma.
Proof
The proof is based on the notion of a controlled vector field. Let be a system of tubular neighborhoods in of strata in where is the associated projection and given by the square norm on each fiber of. By definition, a controlled vector field is a family of vector fields on the strata such that: for each stratum A, there exists a neighborhood of in such that for any,on.
Assume the system is compatible with the map . Then there are two key results due to Thom:
- Given a vector field on N, there exists a controlled vector field on S that is a lift of it:.
- A controlled vector field has a continuous flow.
by
It is a map over and is a homeomorphism since is the inverse. Since the flows preserve the strata, also preserves the strata.