# A Combinatorial Characterization of Geometric 3-Manifolds

## Location

383-N

Tuesday, October 12, 2021 4:00 PM

Daryl Cooper (UC Santa Barbara)

Abstract: Given a compact surface, F, and K in {0,1,-1}, the following are equivalent.

- There is a metric of constant curvature K on F.
- There is an immersion of F into the connect sum of (2-K) projective planes.
- There is a triangulation of F such that every vertex has degree (6-K).

We prove similar statements for 3-manifolds, and each of the eight Thurston geometries.

This gives a new algorithm to find a geometric decomposition.

Joint with Priyam Patel.