Graph manifolds have virtually positive Seifert volume

TL;DR

This paper proves that all closed non-trivial graph manifolds have virtually positive Seifert volume, implying finiteness of certain mapping degree sets unless the target manifold is a special type.

Contribution

It establishes the virtually positive Seifert volume for all closed non-trivial graph manifolds, a new result in 3-manifold topology.

Findings

Graph manifolds have virtually positive Seifert volume.

Finiteness of mapping degree sets for most 3-manifolds.

Special cases where the set of mapping degrees is infinite.

Abstract

This paper shows that the Seifert volume of each closed non-trivial graph manifold is virtually positive. As a consequence, for each closed orientable prime 3-manifold $N$, the set of mapping degrees $\c{D}(M,N)$ is finite for any 3-manifold $M$, unless $N$ is finitely covered by either a torus bundle, or a trivial circle bundle, or the 3-sphere.