Whitney Stratifications

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:

Theorem: (Ehresmann)
Let M\rightarrow{f}N be a proper submersion of manifolds. Then f is locally a trivial fibration.

The idea is that if we start at any point p in N and choose a vector v\in T_{p}N, then we can lift this vector to a vector field on the fiber M_p. We can then use Picards theorem on the existance of unique solutions to ODE’s to lift an open ball B around p to a neighborhood of M_p which is homeomerphic to M_p\times B. 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 S be some closed subset of a manifold M and let M\rightarrow{f}N be a proper submersion. When is f|_{S}:S\rightarrow N a proper submersion?

As an example where this fails, let M=\mathbb{C}^3 and S be the Whitney cusp as pictured below:

Whitney cusp

We can look at this of a degeneration of a family of nodes to a cusp, parameterized by N=\mathbb{C}. Let f be the projection map from M to N. The restriction to S 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 S, we see that the problem occurs while trying to lift tangent vectors ot the origin of N 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 S 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 (X,Y) with X\subset\overline{Y}\setminus Y, there is a unique way of “wiggling” X into Y. 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 (X,Y) satisfies the Whitney condition (b) at a point y\in Y if the following condition holds. Let \{x_i\} be a sequence of points in X converging to y\in Y and \{y_i\} a sequence in Y, also converging to y. Suppose the lines (x_i,y_i) converge to a line l\in \mathbb{P}T_{y}M and the sequence of tangent spaces T_{x_i}X converges in the appropriate Grassmanian to a plane \tau\subset T_{y}M. Then l\subset\tau.

Before we continue, you may be wondering what it means to talk about the “line” (x_i,y_i). If x_i and y_i 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 \mathbb{P}T_{y}M?

The cleanest way to define this seems to be with a blowup construction. Let B be the blowup of M\times M are the diagonal \Delta. Then, B is naturally the disjoint union of M\times M \setminus \Delta and the exceptional divisor E\cong\mathbb{P}TM. Then, the when we say that the lines (x_i,y_i) converge to a line l\in \mathbb{P}T_{y}M, we mean that the sequence of points \{(x_i,y_i)\}\in B converges to the point l\in E.

Definition:
A Whitney stratification of a subset S of a manifold M is a cover of S by pairwise disjoint smooth manifolds \mathcal{S}=\{X_i\} (the strata) such that:

  • locally finite Each point of M meets only finitely strata.
  • frontier condition For each stratum X, it’s frontier \overline{X}\setminus X is a union of strata.
  • Whitney condition Each pair of strata (X,Y) satisfy the Whitney condition (b) at all points of Y.

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 S by the pair (X,Y) where Y is the line running through the nodal and cuspidal singularities and X=S\setminus Y. We will see that this pair is not a Whitney stratification.

Consider the sequence of points \{x_i\} in X drawn as blue dots in the image below, the sequence \{y_i\} in Y drawn as red points and the black point y\in Y.

whitney_cusp_lines1

Clearly, both sequences converge to y. In addition, the black lines show the lines (x_i,y_i). However, the limit of the tangent spaces of the points \{x_i\}, whose cross sections are the thin blue lines in the picture, and the limit of the black lines (x_i,y_i) are orthogonal. Thus, the Whitney condition (b) is not satisfied at y.

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 X to have neighborhood in S 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 \mathcal{S} and a pair of strata Y\subset\overline{X}\setminus X, Y has a neighborhood in X which can be retracted in a controlled way onto Y.

We can now state Thom’s generalization of the Ehresmann fibration theorem.

Theorem: (Thom’s First Isotopy Lemma)
Let M\rightarrow{f}N be a map of smooth manifolds and let S\subset M be a subset of M which admits a Whitney stratification \mathcal{S}. Furthermore, suppose that for each stratum X\in\mathcal{S}, f|_{X}:X\rightarrow N is a proper submersion. Then f|_{S}:S\rightarrow N 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 S embedded in M=\mathbb{C}^3, together with the projection M\rightarrow{f}N=\mathbb{C}. As before, we think of S as a family of curves parameterized by N. In this case, we saw that f|_{S}:S\rightarrow N is not a local fibration.

Suppose we tried to construct a Whitney stratification of S. One way to do this would be to take X_1 as the cuspidal singularity, X_2 as the cusp minus X_1, and X_3 as the rest of S. In this case, the map f|_{X_2}:X_2\rightarrow N 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 S, one of the strata will not submerge onto N.

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1 Response to Whitney Stratifications

  1. alexyoucis says:

    This was very interesting!

    First, some typos:

    -In Ehresmann’s theorem, I think you want M\xrightarrow{f}N, yeah?
    -“existance”—should be—>”existence”
    – “apply these ideas so possibly”—should be—>”apply these ideas to possibly”
    – “And let M\to f N“—should be (maybe?)—> M\xrightarrow{f}N
    – “vectors ot the origin”—-should be—->”vectors at the origin”
    – “does this mean for two points on a manifold and in” ——should be (for clarity)—–>”does this mean for two arbitrary points on a manifold and in”
    -“Let B be the blowup…are the diagonal…”—-should be (I think?)—>”Let B be the blowup…along the diagonal…”
    -“In addition the black lines show…” is kind of at a random place. This should be included before the picture, with the other annotations. Where it is right now confused me.
    – In the statement of Thom’s Isotropy theorem, you may have that M\xrightarrow{f}N issue again.

    Questions:

    – You say lift a vector v\in T_p N to a ‘vector field’ on the fiber M_p. Maybe I’m being silly, but how does one do this? The most obvious thing to do would be to assign to x\in M_p a vector w\in T_x M which maps to v. But, this really doesn’t seen likely to be smooth. Any hints?
    -Also, what are you applying Picard’s theorem to? Are you applying it to the lifted vector field?
    -That Whitney cusp is a great example! Do you know equations for it?
    – So you say that what goes wrong with the Ehresmann argument is just that you can’t “smoothly wiggle tangent vectors”. What literally goes wrong? Can you not lift the tangent vector (because it’s not smooth), or does Picard’s theorem fail?
    – Does your disjoint union need to be of the form \{X_i\} where, like, each (X_i,X_j) is a pair like your (X,Y)? It’s just not clear where this pair is coming from. (Although, I guess this is answered in your definition of Whitney stratification..)
    -With regards to the Thom Isotropy theorem, can you ever use it in the opposite way than you did in the following paragraph? You used there the fact that since there is this subset whose restriction is not a locally trivial fibration to show that it has no Whitney stratification whose restrictions are proper smooth. Could you ever imagine showing that there is no Whitney stratification whose restrictions are proper to show it has no locally trivial fibration?

    Also, is there a good example of a map f:M\to N which is not proper smooth, but for which S=M admits a Whitney stratification whose restrictions are smooth proper (so that f still have to be a locally trivial fibration)?

    Like

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