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.

Template:Harv gives a sketch of the proof. Template:Harv 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 $\ker (d(f{|}_X{)}_{x_i})$ converging to a plane $\tau$ in the Grassmannian, we have $\ker (d(f{|}_Y{)}_y)\subset \tau .$

Let $S\subset M,S^{\prime }\subset N$ be Whitney stratified closed subsets and $p:S\to Z,q:S^{\prime }\to Z$ maps to some smooth manifold Z such that $f:S\to S^{\prime }$ is a map over Z; i.e., $f(S)\subset S^{\prime }$ 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 $S^{\prime }$.
• For each stratum X of S, $f(X)$ lies in a stratum Y of $S^{\prime }$ 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 $f\circ h_1=h_2\circ (f{|}_{p^{-1}(z)}\times \mathrm{id})$.

See also

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References

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