The Hopf Invariant and Simplex Straightening

Let M be a closed oriented 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 . Let X be a closed oriented hyperbolic 3-manifold with injectivity radius less than ǫ at one point. If there is a degree non-zero map from M to X, then we prove that ǫ is at least C . In the 1970’s, Thurston invented simplex straightening. Milnor and Thurston used simplex straightening to bound the degrees of maps to hyperbolic manifolds. In this paper, we extend their method to estimate the Hopf invariant instead of the degree. Here is a degree estimate that Milnor and Thurston proved. Theorem. (Milnor, Thurston) Let M be a closed oriented n-manifold that can be triangulated by N simplices, and let X be a closed oriented hyperbolic n-manifold with volume V. If f is any continuous map from M to X, the norm of the degree of f is bounded by CN/V . We prove a similar estimate for the Hopf invariant. Theorem 1. Let M be a closed oriented 3-manifold which can be triangulated with N simplices. Let f be a continuous map from M to a closed oriented surface of genus 2. If the Hopf invariant of f is defined, then its norm is less than C . (The Hopf invariant can be defined for a map from an oriented 3-manifold to an oriented surface provided that the pullback of the fundamental cohomology class of the surface is zero. See Section 1 for more details.) Our bound for the Hopf invariant grows exponentially in N . We will construct examples where the Hopf invariant is greater than (1+ c) for a universal constant c > 0. Using estimates related to Theorem 1, we give a new proof of a theorem of Soma: Theorem. (Soma) If M is a closed oriented 3-manifold, then there are only finitely many closed oriented hyperbolic 3-manifolds admitting maps of non-zero degree from M. Soma’s theorem follows from the following quantitative estimate: Theorem 2. Let M be a closed oriented 3-manifold that can be triangulated by N simplices. Let X be a closed oriented hyperbolic 3-manifold with injectivity radius ǫ. If there is a map of non-zero degree from M to X, then ǫ is at least C . Roughly speaking, Theorem 2 says that a closed oriented hyperbolic manifold with small injectivity radius at one point must be topologically complicated. Trying to understand Soma’s theorem was the main motivation for the work in this paper. Here is some context for Soma’s theorem. In dimension greater than 3,