Bounded Cohomology and $l_1$-Homology of Three-Manifolds

TL;DR

This paper introduces a new $l_1$-homology class for aspherical 3-manifolds with torus splittings, linking it to Gromov's simplicial volume, and uses it to characterize degree maps and homeomorphisms.

Contribution

It defines a fundamental $l_1$-class for 3-manifolds with torus splittings and relates it to simplicial volume for classifying degree maps and homeomorphisms.

Findings

Defined a 2D fundamental $l_1$-class for 3-manifolds with torus splittings.

Characterized degree maps homotopic to coverings using $l_1$-norm and simplicial volume.

Provided criteria for degree-one maps to be homotopic to homeomorphisms.

Abstract

In this paper we define, for each aspherical orientable 3-manifold $M$ endowed with a \emph{torus splitting} $\c{T}$, a 2-dimensional fundamental $l_1$-class $[M]^{\c{T}}$ whose $l_1$-norm has similar properties as the Gromov simplicial volume of $M$ (additivity under torus splittings and isometry under finite covering maps). Next, we use the Gromov simplicial volume of $M$ and the $l_1$-norm of $[M]^{\c{T}}$ to give a complete characterization of those nonzero degree maps $f\co M\to N$ which are homotopic to a ${\rm deg}(f)$-covering map. As an application we characterize those degree one maps $f\co M\to N$ which are homotopic to a homeomorphism in terms of bounded cohomology classes.