Hyperbolic 4-manifolds with perfect circle-valued Morse functions

TL;DR

This paper constructs specific hyperbolic 4-manifolds with perfect circle-valued Morse functions, demonstrating the existence of manifolds with controlled handle decompositions and bounded topological invariants.

Contribution

It provides explicit examples of hyperbolic 4-manifolds with perfect circle-valued Morse functions and shows that such manifolds can have bounded Betti numbers and handle decompositions.

Findings

Existence of hyperbolic 4-manifolds with perfect circle-valued Morse functions

Construction of a manifold where all generic circle-valued functions are homotopic to a perfect one

Infinite families of manifolds with bounded Betti numbers and handle decompositions

Abstract

We exhibit some (compact and cusped) finite-volume hyperbolic four-manifolds M with perfect circle-valued Morse functions, that is circle-valued Morse functions $f\colon M \to S^1$ with only index 2 critical points. We construct in particular one example where every generic circle-valued function is homotopic to a perfect one. An immediate consequence is the existence of infinitely many finite-volume (compact and cusped) hyperbolic 4-manifolds $M$ having a handle decomposition with bounded numbers of 1- and 3-handles, so with bounded Betti numbers $b_1(M)$, $b_3(M)$ and rank of $\pi_1(M)$.