The Hopf invariant and simplex straightening

TL;DR

This paper establishes bounds on the Hopf invariant for maps from triangulated 3-manifolds to genus 2 surfaces and relates the injectivity radius of hyperbolic 3-manifolds to the degree of maps from other manifolds.

Contribution

It provides explicit exponential bounds linking the complexity of triangulations to topological invariants and geometric properties of hyperbolic 3-manifolds.

Findings

Hopf invariant is at most exponential in the number of simplices

Injectivity radius is bounded below exponentially in the triangulation size

Degree of maps constrains geometric features of hyperbolic manifolds

Abstract

Let M be a closed 3-manifold which can be triangulated with N simplices. We prove that any map from M to a genus 2 surface has Hopf invariant at most C^N. Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than epsilon at one point. If there is a degree non-zero map from M to X, then we prove that epsilon is at least C^{-N}.