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.[1] Like the first isotopy lemma, the lemma was introduced by René Thom. (Mather 2012) 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 [math]\displaystyle{ f : M \to N }[/math] be a smooth map between smooth manifolds and [math]\displaystyle{ X, Y \subset M }[/math] submanifolds such that [math]\displaystyle{ f|_X, f|_Y }[/math] both have differential of constant rank. Then Thom's condition [math]\displaystyle{ (a_f) }[/math] is said to hold if for each sequence [math]\displaystyle{ x_i }[/math] in X converging to a point y in Y and such that [math]\displaystyle{ \operatorname{ker}(d(f|_{X})_{x_i}) }[/math] converging to a plane [math]\displaystyle{ \tau }[/math] in the Grassmannian, we have [math]\displaystyle{ \operatorname{ker}(d(f|_Y)_y) \subset \tau. }[/math][3]
Let [math]\displaystyle{ S \subset M, S' \subset N }[/math] be Whitney stratified closed subsets and [math]\displaystyle{ p : S \to Z, q : S' \to Z }[/math] maps to some smooth manifold Z such that [math]\displaystyle{ f : S \to S' }[/math] is a map over Z; i.e., [math]\displaystyle{ f(S) \subset S' }[/math] and [math]\displaystyle{ q \circ f|_S = p }[/math]. Then [math]\displaystyle{ f }[/math] is called a Thom mapping if the following conditions hold:[3]
- [math]\displaystyle{ f|_S, q }[/math] are proper.
- [math]\displaystyle{ q }[/math] is a submersion on each stratum of [math]\displaystyle{ S' }[/math].
- For each stratum X of S, [math]\displaystyle{ f(X) }[/math] lies in a stratum Y of [math]\displaystyle{ S' }[/math] and [math]\displaystyle{ f : X \to Y }[/math] is a submersion.
- Thom's condition [math]\displaystyle{ (a_f) }[/math] holds for each pair of strata of [math]\displaystyle{ S }[/math].
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 [math]\displaystyle{ h_1 : p^{-1}(z) \times U \to p^{-1}(U), h_2 : q^{-1}(z) \times U \to q^{-1}(U) }[/math] over U such that [math]\displaystyle{ f \circ h_1 = h_2 \circ (f|_{p^{-1}(z)} \times \operatorname{id}) }[/math].[3]
See also
References
- ↑ Mather 2012, Proposition 11.2.
- ↑ § 3 of Bekka, K. (1991). "C-Régularité et trivialité topologique" (in en). Singularity Theory and Its Applications. Lecture Notes in Mathematics (Springer) 1462: 42–62. doi:10.1007/BFb0086373. ISBN 978-3-540-53737-3. https://link.springer.com/chapter/10.1007/BFb0086373.
- ↑ 3.0 3.1 3.2 Mather 2012, § 11.
- 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) (in en). Stratified Mappings - Structure and Triangulability. Lecture Notes in Mathematics. 1102. Springer. doi:10.1007/BFb0101672. ISBN 978-3-540-13898-3. https://link.springer.com/book/10.1007/BFb0101672.
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