In mathematics, especially in differential topology, Thom's first isotopy lemma states: given a smooth map
a closed Whitney stratified subset, if
is a submersion for each stratum
is a locally trivial fibration.
[1] The lemma was originally introduced by René Thom who considered the case when
[2] In that case, the lemma constructs an isotopy from the fiber
The local trivializations that the lemma provide preserve the strata.
On the other hand, it is possible that local trivializations are semialgebraic if the input data is semialgebraic.
[3][4] 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 Bekka's condition (C), which is weaker than Whitney's condition (B).
[5] (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.
The proof[1] is based on the notion of a controlled vector field.
be a system of tubular neighborhoods
given by the square norm on each fiber of
(The construction of such a system relies on the Whitney conditions or something weaker.)
By definition, a controlled vector field is a family of vector fields (smooth of some class)
such that: for each stratum A, there exists a neighborhood
Assume the system
is compatible with the map
(such a system exists).
Then there are two key results due to Thom: The lemma now follows in a straightforward fashion.
Since the statement is local, assume
the coordinate vector fields on
Then, by the lifting result, we find controlled vector fields
Then define by It is a map over
preserve the strata,
also preserves the strata.
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