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]

Thom mapping

Let 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 converging to a plane in the Grassmannian, we have [3]

Let be Whitney stratified closed subsets and maps to some smooth manifold Z such that is a map over Z; i.e., and . Then is called a Thom mapping if the following conditions hold:[3]

  • are proper.
  • is a submersion on each stratum of .
  • For each stratum X of S, lies in a stratum Y of and is a submersion.
  • Thom's condition holds for each pair of strata of .

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 over U such that .[3]

See also

  • Thom–Mather stratified space – topological space equipped with a filtration such that the qutoients (“strata”) are sufficiently manifold-like
  • Thom's first isotopy lemma – Theorem

References

  1. Mather 2012, Proposition 11.2.
  2. § 3 of Bekka, K. (1991). "C-Régularité et trivialité topologique". Singularity Theory and Its Applications. Lecture Notes in Mathematics. Springer. 1462: 42–62. doi:10.1007/BFb0086373. ISBN 978-3-540-53737-3.
  3. 1 2 3 Mather 2012, § 11.
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