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} } .

Source: Wikipedia — Thom's first isotopy lemma (CC BY-SA 4.0)

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} } .

Source: Wikipedia "Thom's first isotopy lemma" · CC BY-SA 4.0

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