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- In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map between smooth manifolds and a closed Whitney stratified subset, if is proper and is a submersion for each stratum of , then is a locally trivial fibration. The lemma was originally introduced by René Thom who considered the case when . In that case, the lemma constructs an isotopy from the fiber to ; whence the name "isotopy lemma". The local trivializations that the lemma provide preserve the strata. However, they are generally not smooth (not even ). On the other hand, it is possible that local trivializations are semialgebraic if the input data is semialgebraic. The lemma is also valid for a more general stratified space such as a stratified space in the sense of Mather but still with the Whitney conditions (or some other conditions). The lemma is also valid for the stratification that satisfies , which is weaker than Whitney's condition (B). (The significance of this is that the consequences of the first isotopy lemma cannot imply Whitney’s condition (B).) Thom's second isotopy lemma is a family version of the first isotopy lemma. (en)
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- 5869 (xsd:nonNegativeInteger)
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- In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map between smooth manifolds and a closed Whitney stratified subset, if is proper and is a submersion for each stratum of , then is a locally trivial fibration. The lemma was originally introduced by René Thom who considered the case when . In that case, the lemma constructs an isotopy from the fiber to ; whence the name "isotopy lemma". Thom's second isotopy lemma is a family version of the first isotopy lemma. (en)
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- Thom's first isotopy lemma (en)
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