Global analysis, analysis on manifolds Generalized smooth structures and supermanifolds Theorem

Thom–Mather isotopy lemmas

Builds on Thom–Mather stratified space Whitney stratification Smooth manifold Submersion Proper map Homeomorphism

Statement

Let $M$ , $N$ and $P$ be smooth manifolds. A map between stratified sets is stratum-preserving if it maps each stratum into a stratum.

(First isotopy lemma.) Let $X \subseteq M$ be a closed subset with a Whitney stratification, and let $f \colon M \to P$ be a smooth map. Assume that $f|_X \colon X \to P$ is proper and that $f|_S \colon S \to P$ is a submersion for every stratum $S$ of $X$ . Then $f|_X$ is a locally trivial fibration with stratum-preserving trivialisations: every $p \in P$ has an open neighbourhood $U$ and a stratum-preserving homeomorphism $h \colon f^{-1}(U) \cap X \to (f^{-1}(p) \cap X) \times U$ with $\mathrm{pr}_U \circ h = f$ .
(Second isotopy lemma.) Let $X \subseteq M$ and $Y \subseteq N$ be closed Whitney stratified subsets, and let $f \colon M \to N$ and $g \colon N \to P$ be smooth maps with $f(X) \subseteq Y$ . Assume that $f|_X$ and $g|_Y$ are proper, and that $f$ maps each stratum of $X$ submersively onto a stratum of $Y$ . Assume that $f|_X$ satisfies Thom's condition $a_f$ , and that $g|_S$ is a submersion for every stratum $S$ of $Y$ . Then $f|_X$ is locally trivial over $P$ , in this sense. Every $p \in P$ has an open neighbourhood $U$ and stratum-preserving homeomorphisms $X \cap (g f)^{-1}(U) \to (X \cap (g f)^{-1}(p)) \times U$ and $Y \cap g^{-1}(U) \to (Y \cap g^{-1}(p)) \times U$ . These homeomorphisms commute with the projections to $U$ and with $f$ .

Remarks

The proof uses the control data of a Thom–Mather stratified space, which Mather constructed on every Whitney stratified set. One lifts the coordinate vector fields of $U$ to controlled vector fields on $X$ , smooth on each stratum and compatible with the tubular projections and distance functions. Controlled vector fields have continuous flows that preserve each stratum, and properness makes the flows defined for the needed times. Integrating them gives the trivialisation. The flows are only continuous across strata, so the trivialisations are homeomorphisms, not diffeomorphisms.

Example: in $M = \R^3$ let $X$ be the union of the cylinder $\set{x^2 + y^2 = 1}$ and the $t$ -axis $\set{x = y = 0}$ , with these two strata, and let $f(x, y, t) = t$ . Each stratum is mapped submersively onto $\R$ , and $f|_X$ is proper since $X \cap \set{\abs{t} \le c}$ is compact. The first lemma gives a homeomorphism of $X$ with $(\text{circle} \sqcup \text{point}) \times \R$ that keeps the two strata apart.

Properness is needed: the inclusion of $M = X = \R \setminus \set{0}$ into $P = \R$ , with one stratum, is a submersion. But its fibres are points over $t \ne 0$ and empty over $0$ , so it is not locally trivial near $0$ .

Going deeper

For a student

For smooth manifolds, Ehresmann's fibration theorem says that a proper submersion is a locally trivial fibration. The first isotopy lemma extends this to Whitney stratified sets, at the price of getting only homeomorphisms. It shows, for instance, that a Whitney stratified set is locally topologically trivial along each stratum. So the local topology near a point stays the same as the point moves within a connected stratum. The second lemma is a key tool in Mather's proof that topologically stable maps are dense. Thom's condition $a_f$ is the hypothesis that makes the two families of trivialisations compatible.

Reading path

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Tag	`7G1PCS2RPG`
URI	`m2://7G1PCS2RPG`
Kind	statement, theorem
Writing	W0 stub
MSC	58A35, 57N80, 32S60
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About	Thom–Mather stratified space, Whitney stratification
Uses	Thom–Mather stratified space, Whitney stratification, Smooth manifold, Submersion, Proper map, Homeomorphism