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.