Complexity functions on 1-dimensional cohomology

TL;DR

This paper introduces a new complexity function on the first cohomology of manifolds and groups, measuring how close classes are to fibrations, with implications for understanding the structure of fibrations and geometric invariants.

Contribution

It defines a novel upper semi-continuous complexity function on cohomology classes, generalizing existing results and linking to the Bieri-Neumann-Strebel invariant.

Findings

Complexity minimisers form open sets, generalizing Tischler's result.

The complexity function is constant on open rays from the origin.

It vanishes exactly on the Bieri-Neumann-Strebel invariant.

Abstract

For a smooth, closed $n$-manifold $M$, we define an upper semi-continuous integer-valued complexity function on $H^1(M;{\mathbb R})$ using Morse theory. This measures how far an integral class is from being a fiber of a fibration. The fact complexity minimisers are open generalises Tischler's result on the openness of classes dual to fibrations. We then use this to define a complexity function on 1-dimensional cohomology of a finitely presented group, which is constant on open rays from the origin and vanishes precisely on the geometric invariant due to Bieri, Neumann and Strebel.