In a perfect world, all “naturally occurring” geometric objects in mathematics and physics would have a nice manifold structure, together with a well-behaved ring of functions. Unfortunately, this is simply not the case. But how do we proceed? Do we part completely from the safety of the power techniques of differential topology, and descend into the wilds of general topology? What do we keep? For me, the answer is the idea of local triviality. A manifold is “locally trivial” in the sense that the local (ambient) topological type of any particular point is that of ordinary Euclidean space.
One of the basic themes of singularity theory is that of stratifying spaces, or chopping up a singular space into a (disjoint) union of smooth or complex manifolds. Not haphazardly, of course. We require that the pieces, or strata, fit together in precise ways for some semblance of regularity. Another basic theme is to consider these singular spaces as embedded inside some ambient smooth or complex manifold, so as to retain as much differential topology as possible, considering strata as (suitably nice!) collections of submanifolds of the ambient space.
First, a partition. Let be a closed subset of a smooth manifold, . This will be our singular space. A (non-empty, duh) collection of submanifolds is a smooth partition of X if:
- is the disjoint union of the ;
- Each is a connected, smooth submanifold of .
- is locally finite. That is, for all , there is an open neighborhood of in such that for only finitely many indices .
Analogously, if is a complex analytic subset of a (connected) complex manifold , where the submanifolds are now required to be connected complex submanifolds of . Additionally, we require that each is an irreducible complex analytic subset of , and that is a complex analytic subset of as well.
Partitions are the basic building block, after strata (in terms of complexity of their definition). What we actually use, however, are stratifications. This additional step serves to partially order the strata of a partition . Precisely, a smooth (resp., complex) partition of is a smooth (resp., complex analytic) stratification if it satisfies the Condition of the Frontier: If () are such that , then and .
So now that has been chopped up into a bunch of nice submanifolds, how exactly do we “piece” these pieces together to understand the geometry and topology of ? That is, how do strata meet each other? The Whitney Conditions (after mathematician Hassler Whitney) are one method of imposing regularity on this “piecing together.” Basically, the Whitney conditions (there are two) control the behavior of “limiting tangent planes” of higher dimensional strata to lower dimensional strata.
Whitney’s condition (a) for a pair of strata (where ) at a point states that, for any sequence of points such that the limit exists (inside the Grassmannian of -planes in the tangent space , where is the ambient manifold), one has the inclusion .
We’d say is a Whitney (a) stratification if every pair of strata satisfies Whitney’s condition (a) at all points .
In terms of conormal spaces to strata, this has a particularly simple expression: the pair satisfies Whitney’s condition (a) at if there is an inclusion of fibers over .
Whitney’s condition (b) for a pair at a point states that, (after fixing a local coordinate system) for all sequences of points such that the limit exists, for all sequences of points such that the limiting “secant line” exists (remember, we fixed a local coordinate system ahead of time, so this makes sense), there is an inclusion as subspaces of . We’d say is a Whitney (b) stratification if this condition holds for all pairs of incident strata.
Thankfully, it doesn’t matter what local coordinate system you pick at the beginning, the condition is independent of that choice. We pick one for the sole purpose of making sense of these “secant lines.” Also, condition (b) implies condition (a), so it’s a strictly stronger requirement. Henceforth, a Whitney stratification will mean a (complex analytic) Whitney (b) stratification.
The main purpose (the only one I’ve seen or cared about, so far) for introducing Whitney stratifications is that the local, ambient topological type of is locally trivial along strata. This is intimately related to the locally cone-like nature of complex (and real!) analytic sets that I (briefly) mention in this post https://brainhelper.wordpress.com/2013/09/26/the-milnor-fibration-and-why-you-should-care/. It’s not exactly “locally Euclidean,” but it’s something!
But, I’m tired, and don’t feel like talking anymore. Until next time.