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

Davis, Michael W.; Januszkiewicz, Tadeusz Hyperbolization of polyhedra. J. Differential Geom. 34 (1991), no. 2, 347—388.

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 with a map
• Any simplicial complex with a nondegenerate (edge-non-collapsing) map (if having a map to seems like a bother, note that the barycentric subdivision comes with a map to for free).

Let for a subcomplex of ; we think of this as decomposing into pieces resembling a simplex.

Now the construction is easy: replace each simplex in with a corresponding piece of . Or more formally, build the fiber product of and over ; this fiber product is denoted by in the paper. From this, we get a natural map .

The vague upshot is this: features of translate into features of , while nonetheless preserving features of . Here are a couple of examples of how assumptions on lead to consequence for .

• If is path-connected, and for each codimension 1 face of , we have , then is a surjection.
• If and are PL-manifolds, and , and , then is a PL-manifold.