Thom's second isotopy lemma

In mathematics, especially in differential topology, Thom's second isotopy lemma is a family version of Thom's first isotopy lemma; i.e., it states a family of maps between Whitney stratified spaces is locally trivial when it is a Thom mapping. Like the first isotopy lemma, the lemma was introduced by René Thom.
gives a sketch of the proof. gives a simplified proof. Like the first isotopy lemma, the lemma also holds for the stratification with Bekka's condition (C), which is weaker than Whitney's condition.

Thom mapping

Let be a smooth map between smooth manifolds and submanifolds such that both have differential of constant rank. Then Thom's condition is said to hold if for each sequence in X converging to a point y in Y and such that converging to a plane in the Grassmannian, we have
Let be Whitney stratified closed subsets and maps to some smooth manifold Z such that is a map over Z; i.e., and. Then is called a Thom mapping if the following conditions hold:

are proper.
is a submersion on each stratum of.
For each stratum X of S, lies in a stratum Y of and is a submersion.
Thom's condition holds for each pair of strata of.

Then Thom's second isotopy lemma says that a Thom mapping is locally trivial over Z; i.e., each point z of Z has a neighborhood U with homeomorphisms over U such that.