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 formula_1 be a smooth map between smooth manifolds and formula_2 submanifolds such that formula_3 both have differential of constant rank. Then Thom's condition formula_4 is said to hold if for each sequence formula_5 in "X" converging to a point "y" in "Y" and such that formula_6 converging to a plane formula_7 in the Grassmannian, we have formula_8
Let formula_9 be Whitney stratified closed subsets and formula_10 maps to some smooth manifold "Z" such that formula_11 is a map over "Z"; i.e., formula_12 and formula_13. Then formula_14 is called a Thom mapping if the following conditions hold:
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 formula_23 over "U" such that formula_24.
References.
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