Thom's second isotopy lemma explained

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.

gives a sketch of the proof. 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).^[1]

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) x U \to p ^{-1} (U), h _{2} : q ^{-1} (z) x U \to q ^{-1} (U) over U such that f \circ h _{1} = h _{2} \circ (f| _{p ^{-1} (z)} x \operatorname{id}) .

References

• 10.1090/S0273-0979-2012-01383-6. Notes on Topological Stability . 2012 . Mather . John . Bulletin of the American Mathematical Society . 49 . 4 . 475–506 . free .
• 10.1090/S0002-9904-1969-12138-5. Ensembles et morphismes stratifiés . 1969 . Thom . R. . Bulletin of the American Mathematical Society . 75 . 2 . 240–284 . free .
• Book: Verona . Andrei . Stratified Mappings - Structure and Triangulability . Lecture Notes in Mathematics . 1984 . 1102 . Springer . 10.1007/BFb0101672 . 978-3-540-13898-3 . en.

Notes and References

1. § 3 of Bekka . K. . C-Régularité et trivialité topologique . Singularity Theory and Its Applications . Lecture Notes in Mathematics . 1991 . 1462 . 42–62 . 10.1007/BFb0086373 . Springer . 978-3-540-53737-3 . en.