Three-dimensional manifolds with poor spines

TL;DR

This paper studies special spines of three-manifolds, showing that manifolds with boundary have finitely many poor spines with uniform true vertices, and relates spine complexity to hyperbolic manifolds with geodesic boundary.

Contribution

It establishes finiteness and uniformity of poor special spines for manifolds with boundary and links spine complexity to hyperbolic geometry with totally geodesic boundary.

Findings

Finite number of poor special spines for manifolds with boundary.

All poor spines of a given manifold have the same number of true vertices.

Complexity of certain hyperbolic manifolds equals the number of true vertices.

Abstract

A special spine of a three-manifold is said to be poor if it does not contain proper simple subpolyhedra. Using the Turaev-Viro invariants, we establish that every compact three-dimensional manifold M with connected nonempty boundary has a finite number of poor special spines. Moreover, all poor special spines of the manifold M have the same number of true vertices. We prove that the complexity of a compact hyperbolic three-dimensional manifold with totally geodesic boundary that has a poor special spine with two 2-components and n true vertices is n. Such manifolds are constructed for infinitely many values of n.