Whitney stratifications

Question

Many results on characteristic classes of singular varieties (as well as other singularity-theoretic constructions) make use of a so-called "Whitney stratification" of the variety under consideration, which from what I understand is a filtration of the variety into smooth locally closed subvarieties subject to some technical conditions (the technical conditions may be found here: https://en.wikipedia.org/wiki/Whitney_conditions). However, other than merely reading the bare-bones definitions of these technical conditions, I have never seen these technical conditions explicitly used. Moreover, I have never once seen a concrete example of a Whitney stratification. As such, an elucidation of the technical conditions and why they are useful along with a concrete example of a Whitney stratification (perhaps say a Whitney umbrella) would be greatly appreciated. Thanks!

Answer 1

I struggled with the Whitney conditions myself. More precisely, I wanted to understand the geometric significance of these conditions. Apparently Whitney was seeking a simple way to characterize equisingularity: along a connected component of stratum the stratification looks the same. Technically the Whitney conditions are local conditions that guarantee a desirable global property of the stratification: normal equisingularity.

This turned out to be a difficult problem, first solved by R. Thom, but the first account I could understand would be that by J. Mather.

You can read more about this in Section 4.2 of my book An Invitation to Morse Theory, 2nd edition. There you will find pictures, examples, and the basic geometric consequences of the Whitney conditions, including normal equisingularity. All that in 12 pages and no technical proofs, though rather generous references.

You can find a (less efficient) precursor of Section 4.2 here.

Update In Section 4.3 of the same book I prove that the stratification of a compact manifold by the unstable manifolds of a gradient flow is Whitney iff the flow satisfies the Smale transversality conditions. In particular, the Schubert stratification of Grassmanians is Whitney. For more examples I refer to chapter 7 of this paper.

Answer 2

Here is an example:

Let $V = \lbrace y^2 = t^2 x^2 + x^3\rbrace \subset \mathbb R^3$. Then the singular set of $V$ is the whole $t$-axis.

Let $Y$ be the $t$-axis and $X = V - Y$.

Now set $X_1 = X \cap \lbrace x > 0 \rbrace$,

$X_2 = X \cap \lbrace x < 0 \rbrace \cap \lbrace t>0 \rbrace$, and

$X_3 = X \cap \lbrace x < 0 \rbrace \cap \lbrace t<0 \rbrace$.

I wish I could draw pictures but I don't know how to do that here. It's fairly easy though.

Notice that, $X_1$, $X_2$, $X_3$ and $Y$ are smooth submanifolds of $\mathbb R^3$, they are disjoint, and their union is $V$. Thus we have stratified $V$ into submanifolds.

You can now verify that this stratification of $V$ is an $(a)$-regular stratification. However, it is not a $(b)$-regular (Whitney) stratification. Verify that $X_2$ and $X_3$ are not $(b)$-regular over $Y$.

Let $Z$ be the origin and $W = Y - Z$. Now, $X_1$, $X_2$, $X_3$, $W$ and $Z$ are disjoint, smooth submanifolds of $\mathbb R^3$ and their union is $V$. This stratification is $(b)$-regular or what is called a Whitney stratification of $V$.

I hope the example helps visualizing $(a)$-regular and $(b)$-regular stratifications.

You can read `Notes on topological stability' by Mather for properties of Whitney conditions (yes it is published now after more than 40 years).

https://www.ams.org/journals/bull/2012-49-04/S0273-0979-2012-01383-6/S0273-0979-2012-01383-6.pdf

Another good reference is the Ph.D. thesis of David Trotman titled `Whitney stratifications: faults and detectors'.