Stratifications have been coming up a lot recently, and I finally decided to read Mather’s Notes on Topological Stability in the hope of obtaining a more concrete understanding.
In this post I’ll try to give a motivation for Whitney stratifications, and an overview of the key insights.
In the world of manifolds, our absolute understanding of the local picture frequently allows us to translate differential information into topological information. A basic example is the inverse function theorem and it’s many corollaries. As a concrete example, consider Ehresmann’s fibration lemma:
Let be a proper submersion of manifolds. Then is locally a trivial fibration.
The idea is that if we start at any point in and choose a vector , then we can lift this vector to a vector field on the fiber . We can then use Picards theorem on the existance of unique solutions to ODE’s to lift an open ball around to a neighborhood of which is homeomerphic to . I will refer to this as the “Ehresmann argument”.
Now, the question is how to generalize these ideas so possibly singular subsets of a manifold. For example, let be some closed subset of a manifold and let be a proper submersion. When is a proper submersion?
As an example where this fails, let and be the Whitney cusp as pictured below:
We can look at this of a degeneration of a family of nodes to a cusp, parameterized by . Let be the projection map from to . The restriction to is clearly not locally a fibration since one of the fibers is a cuspidal curve while the others are nodal curves.
If we try to run the “Ehresmann argument” on , we see that the problem occurs while trying to lift tangent vectors ot the origin of to the fiber which is a cusp. Another way of saying this is that there is no smooth way to wiggle the cusp around in the family.
Whitney tries to circumvent this with the following idea. First, break up into a disjoint union of smooth manifolds (strata). On each of these manifolds, we know that the Ehresmann argument goes through. The difficulty now lies in the way that these manifolds lie in relation to one another. Inspired by the above, we would like some condition that would ensure that for each pair of strata with , there is a unique way of “wiggling” into . Because if we had this, then we would make local arguments on each strata and then glue them together by relying on the above property.
The condition that Whitney came up with is the following. It is called “Whitney condition (b)”, and it implies a simpler one called “Whitney condition (a)”.
Whitney Condition (b): We say that a pair satisfies the Whitney condition (b) at a point if the following condition holds. Let be a sequence of points in converging to and a sequence in , also converging to . Suppose the lines converge to a line and the sequence of tangent spaces converges in the appropriate Grassmanian to a plane . Then .
Before we continue, you may be wondering what it means to talk about the “line” . If and are in a vector space then this is just the line spanned by the points. But what does this mean for two points on a manifold and in what sense does it converge to a line in ?
The cleanest way to define this seems to be with a blowup construction. Let be the blowup of are the diagonal . Then, is naturally the disjoint union of and the exceptional divisor . Then, the when we say that the lines converge to a line , we mean that the sequence of points converges to the point .
A Whitney stratification of a subset of a manifold is a cover of by pairwise disjoint smooth manifolds (the strata) such that:
- locally finite Each point of meets only finitely strata.
- frontier condition For each stratum , it’s frontier is a union of strata.
- Whitney condition Each pair of strata satisfy the Whitney condition (b) at all points of .
Now, what is the motivation for such a definition? One way to see it is to look at where it fails for the Whitney cusp. Suppose we tried to stratify the Whitney cusp by the pair where is the line running through the nodal and cuspidal singularities and . We will see that this pair is not a Whitney stratification.
Consider the sequence of points in drawn as blue dots in the image below, the sequence in drawn as red points and the black point .
Clearly, both sequences converge to . In addition, the black lines show the lines . However, the limit of the tangent spaces of the points , whose cross sections are the thin blue lines in the picture, and the limit of the black lines are orthogonal. Thus, the Whitney condition (b) is not satisfied at .
With a little imagination, we can see that the “collapse of this black line” is a symptom of the inability to wiggle the cupidal curve inside our family of curves.
With this in mind, the intuition behind the Whitney condition is that if forces a strata to have neighborhood in which is nice enough to allow us to move it freely. This can be stated formally in terms of what Mather calls “control data”, which, roughly speaking, says that given a Whitney stratification and a pair of strata , has a neighborhood in which can be retracted in a controlled way onto .
We can now state Thom’s generalization of the Ehresmann fibration theorem.
Theorem: (Thom’s First Isotopy Lemma)
Let be a map of smooth manifolds and let be a subset of which admits a Whitney stratification . Furthermore, suppose that for each stratum , is a proper submersion. Then is a local fibration.
In order to get a feel for this theorem, let’s go back to the example we started with of Whitney cusp embedded in , together with the projection . As before, we think of as a family of curves parameterized by . In this case, we saw that is not a local fibration.
Suppose we tried to construct a Whitney stratification of . One way to do this would be to take as the cuspidal singularity, as the cusp minus , and as the rest of . In this case, the map is clearly not a submersion, so the hypotheses of Thom’s lemma do not hold. In fact, we see by the lemma that for any Whitney stratification of , one of the strata will not submerge onto .