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.

Published by brianhepler

I'm a third-year math postdoc at the University of Wisconsin-Madison, where I work as a member of the geometry and topology research group. Generally speaking, I think math is pretty neat; and, if you give me the chance, I'll talk your ear off. 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 to show you a world of pure imagination. View more posts

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