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 → N {\displaystyle f:M\to N} be a smooth map between smooth manifolds and X , Y ⊂ M {\displaystyle X,Y\subset M} submanifolds such that f | X , f | Y {\displaystyle f|_{X},f|_{Y}} both have differential of constant rank. Then Thom's condition ( a f ) {\displaystyle (a_{f})} is said to hold if for each sequence x i {\displaystyle x_{i}} in X converging to a point y in Y and such that ker ( d ( f | X ) x i ) {\displaystyle \operatorname {ker} (d(f|_{X})_{x_{i}})} converging to a plane τ {\displaystyle \tau } in the Grassmannian, we have ker ( d ( f | Y ) y ) ⊂ τ . {\displaystyle \operatorname {ker} (d(f|_{Y})_{y})\subset \tau .}
Let S ⊂ M , S ′ ⊂ N {\displaystyle S\subset M,S'\subset N} be Whitney stratified closed subsets and p : S → Z , q : S ′ → Z {\displaystyle p:S\to Z,q:S'\to Z} maps to some smooth manifold Z such that f : S → S ′ {\displaystyle f:S\to S'} is a map over Z; i.e., f ( S ) ⊂ S ′ {\displaystyle f(S)\subset S'} and q ∘ f | S = p {\displaystyle q\circ f|_{S}=p} . Then f {\displaystyle f} is called a Thom mapping if the following conditions hold:
f | S , q {\displaystyle f|_{S},q} are proper.
q {\displaystyle q} is a submersion on each stratum of S ′ {\displaystyle S'} . For each stratum X of S, f ( X ) {\displaystyle f(X)} lies in a stratum Y of S ′ {\displaystyle S'} and f : X → Y {\displaystyle f:X\to Y} is a submersion. Thom's condition ( a f ) {\displaystyle (a_{f})} holds for each pair of strata of S {\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 h 1 : p − 1 ( z ) × U → p − 1 ( U ) , h 2 : q − 1 ( z ) × U → q − 1 ( U ) {\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 f ∘ h 1 = h 2 ∘ ( f | p − 1 ( z ) × id ) {\displaystyle f\circ h_{1}=h_{2}\circ (f|_{p^{-1}(z)}\times \operatorname {id} )} .
See also Thom–Mather stratified space – Way of decomposing a topological space Thom's first isotopy lemma – TheoremPages displaying short descriptions with no spaces
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