TITLE: Whitney stratifications
QUESTION [14 upvotes]: 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 techinical conditions (the technical conditions may be found here: http://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!

REPLY [11 votes]: 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).
http://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'.