Local Triviality

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 $X$ be a closed subset of a smooth manifold, $M$ . This will be our singular space. A (non-empty, duh) collection of submanifolds $\mathfrak{S} = \{S_\alpha\}_\alpha$ is a smooth partition of X if:

$X$ is the disjoint union of the $S_\alpha$ ;
Each $S_\alpha$ is a connected, smooth submanifold of $M$ .
$\mathfrak{S}$ is locally finite. That is, for all $x \in X$ , there is an open neighborhood $U$ of $x$ in $X$ such that $U \cap S_\alpha \neq \emptyset$ for only finitely many indices $\alpha$ .

Analogously, if $X$ is a complex analytic subset of a (connected) complex manifold $M$ , where the submanifolds $S_\alpha$ are now required to be connected complex submanifolds of $M$ . Additionally, we require that each $\overline{S_\alpha}$ is an irreducible complex analytic subset of $M$ , and that $\overline{S_\alpha} \backslash S_\alpha$ is a complex analytic subset of $M$ 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 $\mathfrak{S}$ . Precisely, a smooth (resp., complex) partition $\mathfrak{S}$ of $X$ is a smooth (resp., complex analytic) stratification if it satisfies the Condition of the Frontier: If $S_\alpha, S_\beta \in \mathfrak{S}$ ( $S_\alpha \neq S_\beta$ ) are such that $S_\alpha \cap \overline{S_\beta} \neq \emptyset$ , then $S_\alpha \subset \overline{S_\beta}$ and $\dim S_\alpha < \dim S_\beta$ .

So now that $X$ 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 $X$ ? 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 $(S_\beta,S_\alpha)$ (where $S_\alpha,S_\beta \in \mathfrak{S}, S_\alpha \neq S_\beta$ ) at a point $p \in S_\alpha \cap \overline{S_\beta}$ states that, for any sequence of points $S_\beta \ni p_i \to p \in S_\alpha$ such that the limit $\lim_{i} T_{p_i} S_\beta = \tau$ exists (inside the Grassmannian of $\dim S_\beta$ -planes in the tangent space $T_p M$ , where $M$ is the ambient manifold), one has the inclusion $T_p S_\alpha \subseteq \tau$ .

We’d say $\mathfrak{S}$ is a Whitney (a) stratification if every pair $(S_\beta,S_\alpha)$ of strata satisfies Whitney’s condition (a) at all points $p \in S_\alpha \cap \overline{S_\beta}$ .

In terms of conormal spaces to strata, this has a particularly simple expression: the pair $(S_\beta,S_\alpha)$ satisfies Whitney’s condition (a) at $p \in S_\alpha \cap \overline{S_\beta}$ if there is an inclusion of fibers $\overline{T_{S_\beta,p}^*M} \subseteq T_{S_\alpha,p}^*M$ over $p$ .

Whitney’s condition (b) for a pair $(S_\beta,S_\alpha)$ at a point $p \in S_\alpha \cap \overline{S_\beta}$ states that, (after fixing a local coordinate system) for all sequences of points $S_\beta \ni p_i \to p \in S_\alpha$ such that the limit $\lim_i T_{p_i}S_\beta = \tau$ exists, for all sequences of points $S_\alpha q_i \to p \in S_\alpha$ such that the limiting “secant line” $\lim_i \overline{q_i p_i} = \mathfrak{l}$ exists (remember, we fixed a local coordinate system ahead of time, so this makes sense), there is an inclusion $\mathfrak{l} \subseteq \tau$ as subspaces of $T_pM$ . We’d say $\mathfrak{S}$ 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 $X$ 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.

Local Triviality

Published by brianhepler

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