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.

[1] Like the first isotopy lemma, the lemma was introduced by René Thom.

(Mather 2012, § 11) gives a sketch of the proof.

(Verona 1984) 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).

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

ker ⁡ ( d ( f

converging to a plane

τ

in the Grassmannian, we have

) ⊂ τ .

be Whitney stratified closed subsets and

maps to some smooth manifold Z such that

is a map over Z; i.e.,

is called a Thom mapping if the following conditions hold:[3] 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

× id )

{\displaystyle f\circ h_{1}=h_{2}\circ (f|_{p^{-1}(z)}\times \operatorname {id} )}

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