Local Triviality -> Locally Cone-Like

So, locally cone-like…what did I mean by that?  Let X \subseteq M be a complex analytic subset of a complex manifold, \mathfrak{S} a Whitney stratification of X, p \in X.  Suppose we’ve given M a Riemannian metric, r, and denote by B_\delta(p) = \{ q \in M | r(q,p) \leq \delta \} the “ball off radius \delta about p.” Choosing local coordinates about p in M, we might as well assume that we’re dealing with the ordinary Euclidean distance.  ANYWAY, for sufficiently small \delta > 0, the “boundary” \partial B_\delta(p) transversely intersects all strata of \mathfrak{S} (this isn’t too hard to show…suppose not, use the local finiteness criterion for \mathfrak{S}, and apply the Curve Selection Lemma to each stratum to achieve a contradiction).  Then, there is a homeomorphism (preserving the strata), which I’ll call a \mathfrak{S}-homeomorphism, of germs:

(B_\delta(p) \cap X, p) \overset{\thicksim}{\to} (Cone(\partial B_\delta(p) \cap X), p)

This can be rephrased a bit more efficiently.  Let r: X \to \mathbb{R} be “distance squared from p.”  Then, for \delta > 0 sufficiently small, the map r: X \to [0,\delta] is a proper, stratified submersion.  Think about it.   The “stratified submersion” part tells you that the level sets of r are transverse to strata.  Properness allows us to invoke something called “Thom’s first isotopy lemma,” which tells us the cone bit.  

 

 

Author: brianhepler

I'm a second-year math postdoc at the University of Wisconsin-Madison, and I think math is pretty neat. Especially the more abstract stuff. It's really hard to communicate that love with the general population, but I'm going to do my best.

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