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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