(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 mappingedit
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 alsoedit
Thom–Mather stratified space – topological space equipped with a filtration such that the qutoients (“strata”) are sufficiently manifold-likePages displaying wikidata descriptions as a fallback
^§ 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. ISBN978-3-540-53737-3.
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). Stratified Mappings - Structure and Triangulability. Lecture Notes in Mathematics. Vol. 1102. Springer. doi:10.1007/BFb0101672. ISBN978-3-540-13898-3.
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thom, second, isotopy, lemma, mathematics, especially, differential, topology, family, version, thom, first, isotopy, lemma, states, family, maps, between, whitney, stratified, spaces, locally, trivial, when, thom, mapping, like, first, isotopy, lemma, 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 Rene 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 editLet f M N displaystyle f M to N nbsp be a smooth map between smooth manifolds and X Y M displaystyle X Y subset M nbsp submanifolds such that f X f Y displaystyle f X f Y nbsp both have differential of constant rank Then Thom s condition a f displaystyle a f nbsp is said to hold if for each sequence x i displaystyle x i nbsp 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 nbsp converging to a plane t displaystyle tau nbsp in the Grassmannian we have ker d f Y y t displaystyle operatorname ker d f Y y subset tau nbsp 3 Let S M S N displaystyle S subset M S subset N nbsp be Whitney stratified closed subsets and p S Z q S Z displaystyle p S to Z q S to Z nbsp maps to some smooth manifold Z such that f S S displaystyle f S to S nbsp is a map over Z i e f S S displaystyle f S subset S nbsp and q f S p displaystyle q circ f S p nbsp Then f displaystyle f nbsp is called a Thom mapping if the following conditions hold 3 f S q displaystyle f S q nbsp are proper q displaystyle q nbsp is a submersion on each stratum of S displaystyle S nbsp For each stratum X of S f X displaystyle f X nbsp lies in a stratum Y of S displaystyle S nbsp and f X Y displaystyle f X to Y nbsp is a submersion Thom s condition a f displaystyle a f nbsp holds for each pair of strata of S displaystyle S nbsp 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 nbsp 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 nbsp 3 See also editThom Mather stratified space topological space equipped with a filtration such that the qutoients strata are sufficiently manifold likePages displaying wikidata descriptions as a fallback Thom s first isotopy lemma TheoremPages displaying short descriptions with no spacesReferences edit Mather 2012 Proposition 11 2 3 of Bekka K 1991 C Regularite et trivialite topologique Singularity Theory and Its Applications Lecture Notes in Mathematics Springer 1462 42 62 doi 10 1007 BFb0086373 ISBN 978 3 540 53737 3 a b c 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 stratifies Bulletin of the American Mathematical Society 75 2 240 284 doi 10 1090 S0002 9904 1969 12138 5 Verona Andrei 1984 Stratified Mappings Structure and Triangulability Lecture Notes in Mathematics Vol 1102 Springer doi 10 1007 BFb0101672 ISBN 978 3 540 13898 3 nbsp This topology related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Thom 27s second isotopy lemma amp oldid 1136445551, wikipedia, wiki, book, books, library,