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Thom's second isotopy lemma

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated January 1, 2026

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).

01Thom 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} ).

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This article is adapted from the Wikipedia article “Thom's second isotopy lemma”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Thom-Mather stratified space

In topology, a branch of mathematics, an abstract stratified space, or a Thom-Mather stratified space is a topological space X that has been decomposed into pieces called strata; these strata are manifolds and are required to fit together in a certain way. Thom-Mather stratified spaces provide a purely topological setting for the study of singularities analogous to the more differential-geometric theory of Whitney.

Thom's first isotopy lemma

In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map f : M → N {\displaystyle f:M\to N} between smooth manifolds and S ⊂ M {\displaystyle S\subset M} a closed Whitney stratified subset, if f | S {\displaystyle f|_{S}} is proper and f | A {\displaystyle f|_{A}} is a submersion for each stratum A {\displaystyle A} of S {\displaystyle S} , then f | S {\displaystyle f|_{S}} is a locally trivial fibration. The lemma was originally introduced by René Thom who considered the case when N = R {\displaystyle N=\mathbb {R} } .