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.

(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).

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 $\operatorname {ker} (d(f|_{X})_{x_{i}})$ converging to a plane $\tau$ in the Grassmannian, we have $\operatorname {ker} (d(f|_{Y})_{y})\subset \tau .$

Let $S\subset M,S'\subset N$ be Whitney stratified closed subsets and $p:S\to Z,q:S'\to Z$ maps to some smooth manifold Z such that $f:S\to S'$ is a map over Z; i.e., $f(S)\subset S'$ 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 $S'$.
• For each stratum X of S, $f(X)$ lies in a stratum Y of $S'$ 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 $f\circ h_{1}=h_{2}\circ (f|_{p^{-1}(z)}\times \operatorname {id} )$.

See also

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

References

• Mather, John (2012). "Notes on Topological Stability". Bulletin of the American Mathematical Society. 49 (4): 475–506. doi:10.1090/S0273-0979-2012-01383-6.
• Thom, R. (1969). "Ensembles et morphismes stratifiés". Bulletin of the American Mathematical Society. 75 (2): 240–284. doi:10.1090/S0002-9904-1969-12138-5.
• Verona, Andrei (1984). Stratified Mappings - Structure and Triangulability. Springer.