Thom's first isotopy lemma explained
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.
[1] [2] 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).[3] (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.
Proof
The proof is based on the notion of a controlled vector field. Let
be a system of
tubular neighborhoods
in
of strata
in
where
is the associated projection and
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)
on the strata
such that: for each stratum
A, there exists a neighborhood
of
in
such that for any
,
on
.
Assume the system
is compatible with the map
(such a system exists). Then there are two key results due to Thom:
- Given a vector field
on
N, there exists a controlled vector field
on
S that is a lift of it:
.
- A controlled vector field has a continuous flow (despite the fact that a controlled vector field is discontinuous).
The lemma now follows in a straightforward fashion. Since the statement is local, assume
and
the coordinate vector fields on
. Then, by the lifting result, we find controlled vector fields
on
such that
f*(\widetilde{\partiali})=\partiali\circf
. Let
be the flows associated to them. Then define
by
H(y,t)=\varphin(tn,\phin-1(tn-1, … ,\varphi1(t1,y) … )).
It is a map over
and is a homeomorphism since
G(x)=(\varphi1(-t1, … ,\varphin(-tn,x) … ),t),t=f(x)
is the inverse. Since the flows
preserve the strata,
also preserves the strata.
See also
References
- 10.1090/S0273-0979-2012-01383-6. Notes on Topological Stability . 2012 . Mather . John . Bulletin of the American Mathematical Society . 49 . 4 . 475–506 .
- 10.1090/S0002-9904-1969-12138-5. Ensembles et morphismes stratifiés . 1969 . Thom . R. . Bulletin of the American Mathematical Society . 75 . 2 . 240–284 . free .
External links
- https://mathoverflow.net/questions/23259/thom-first-isotopy-lemma-in-o-minimal-structures
Notes and References
- Book: Real Analytic and Algebraic Geometry: Proceedings of the International Conference, Trento (Italy), September 21-25th, 1992 . 9783110881271 . Broglia . Fabrizio . Galbiati . Margherita . Tognoli . Alberto . 11 July 2011 . Walter de Gruyter .
- Editorial note: in fact, local trivializations can be definable if the input date is definable, according to https://ncatlab.org/toddtrimble/published/Surface+diagrams
- § 3 of Book: Bekka . K. . Singularity Theory and its Applications . C-Régularité et trivialité topologique . Lecture Notes in Mathematics . 1991 . 1462 . 42–62 . 10.1007/BFb0086373 . https://link.springer.com/chapter/10.1007/BFb0086373 . Springer . 978-3-540-53737-3 . en.