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).^[2]

Let ${\displaystyle f:M\to N}$ be a smooth map between smooth manifolds and ${\displaystyle X,Y\subset M}$ submanifolds such that ${\displaystyle f|_{X},f|_{Y}}$ both have differential of constant rank. Then Thom's condition ${\displaystyle (a_{f})}$ is said to hold if for each sequence ${\displaystyle x_{i}}$ in X converging to a point y in Y and such that ${\displaystyle \operatorname {ker} (d(f|_{X})_{x_{i}})}$ converging to a plane ${\displaystyle \tau }$ in the Grassmannian, we have ${\displaystyle \operatorname {ker} (d(f|_{Y})_{y})\subset \tau .}$^[3]

Let ${\displaystyle S\subset M,S'\subset N}$ be Whitney stratified closed subsets and ${\displaystyle p:S\to Z,q:S'\to Z}$ maps to some smooth manifold Z such that ${\displaystyle f:S\to S'}$ is a map over Z; i.e., ${\displaystyle f(S)\subset S'}$ and ${\displaystyle q\circ f|_{S}=p}$. Then ${\displaystyle f}$ is called a Thom mapping if the following conditions hold:^[3]

• ${\displaystyle f|_{S},q}$ are proper.
• ${\displaystyle q}$ is a submersion on each stratum of ${\displaystyle S'}$.
• For each stratum X of S, ${\displaystyle f(X)}$ lies in a stratum Y of ${\displaystyle S'}$ and ${\displaystyle f:X\to Y}$ is a submersion.
• Thom's condition ${\displaystyle (a_{f})}$ holds for each pair of strata of ${\displaystyle S}$.

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 ${\displaystyle h_{1}:p^{-1}(z)\times U\to p^{-1}(U),h_{2}:q^{-1}(z)\times U\to q^{-1}(U)}$ over U such that ${\displaystyle f\circ h_{1}=h_{2}\circ (f|_{p^{-1}(z)}\times \operatorname {id} )}$.^[3]

• Thom–Mather stratified space – Topological space
• Thom's first isotopy lemma – TheoremPages displaying short descriptions with no spaces