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 (B).

Thom mapping
Let $$f : M \to N$$ be a smooth map between smooth manifolds and $$X, Y \subset M$$ submanifolds such that $$f|_X, f|_Y$$ both have differential of constant rank. Then Thom's condition $$(a_f)$$ is said to hold if for each sequence $$x_i$$ in X converging to a point y in Y and such that $$\operatorname{ker}(d(f|_{X})_{x_i})$$ converging to a plane $$\tau$$ in the Grassmannian, we have $$\operatorname{ker}(d(f|_Y)_y) \subset \tau.$$

Let $$S \subset M, S' \subset N$$ be Whitney stratified closed subsets and $$p : S \to Z, q : S' \to Z$$ maps to some smooth manifold Z such that $$f : S \to S'$$ is a map over Z; i.e., $$f(S) \subset S'$$ and $$q \circ f|_S = p$$. Then $$f$$ is called a Thom mapping if the following conditions hold:
 * $$f|_S, q$$ are proper.
 * $$q$$ is a submersion on each stratum of $$S'$$.
 * For each stratum X of S, $$f(X)$$ lies in a stratum Y of $$S'$$ and $$f : X \to Y$$ is a submersion.
 * Thom's condition $$(a_f)$$ holds for each pair of strata of $$S$$.

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 $$h_1 : p^{-1}(z) \times U \to p^{-1}(U), h_2 : q^{-1}(z) \times U \to q^{-1}(U)$$ over U such that $$f \circ h_1 = h_2 \circ (f|_{p^{-1}(z)} \times \operatorname{id})$$.