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).§ 3 of

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 $\operatorname(d(f, _)_)$ converging to a plane $\tau$ in the Grassmannian, we have $\operatorname(d(f, _Y)_y) \subset \tau.$

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 $h_1 : p^(z) \times U \to p^(U), h_2 : q^(z) \times U \to q^(U)$ over ''U'' such that $f \circ h_1 = h_2 \circ (f, _ \times \operatorname)$.

See also

*
*

References

*
*
*

Differential topology
Lemmas
Stratifications

{{topology-stub