Hyperbolization of Polyhedra

 2008-07-26 15:14:56 +0000 talks mathematics

I gave a talk in the Farb and Friends Student Seminar (back in March!) on:


This is an awesome paper—well-worth a few words on every blog!

The construction is way easier than you might think. The ingredients:

Let $X_J = f^{-1}(J)$ for $J$ a subcomplex of $\Delta^n$; we think of this as decomposing $X$ into pieces resembling a simplex.

Now the construction is easy: replace each simplex in $K$ with a corresponding piece of $X$. Or more formally, build the fiber product of $X$ and $|K|$ over $\Delta^n$; this fiber product is denoted by $X \tilde{\Delta} K$ in the paper. From this, we get a natural map $f_K : X \tilde{\Delta} K \to K$.

The vague upshot is this: features of $X$ translate into features of $X \tilde{\Delta} K$, while nonetheless preserving features of $K$. Here are a couple of examples of how assumptions on $X$ lead to consequence for $X \tilde{\Delta} K$.