Hyperbolization of Polyhedra

 July 26, 2008 talks mathematics

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I gave a talk in the Farb and Friends Student Seminar (back in March!) on [1].

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

The construction is way easier than you might think. The ingredients:
  • A model space X with a map f : X \to \Delta^n
  • Any simplicial complex K with a nondegenerate (edge-non-collapsing) map K \to \Delta^n (if having a map to \Delta^n seems like a bother, note that the barycentric subdivision K&39; comes with a map to \Delta^n for free).

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 .
  • If X is path-connected, and for each codimension 1 face \alpha of \Delta^n , we have Error:LaTeX failed:
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  • If X and K are PL-manifolds, and \dim X_J = \dim J , and \partial X_J = X_{\partial J} , then X \tilde{\Delta} K is a PL-manifold.

[1] M.W. Davis, T. Januszkiewicz, Hyperbolization of polyhedra, J. Differential Geom. 34 (1991) 347–388. http://projecteuclid.org/euclid.jdg/1214447212.