Finiteness of nonzero degree maps between three-manifolds

TL;DR

This paper proves that for any nonzero degree, an orientable closed 3-manifold can only map onto finitely many distinct irreducible non-geometric 3-manifolds, establishing finiteness results in 3-manifold mappings.

Contribution

It establishes the finiteness of nonzero degree maps from a fixed 3-manifold onto distinct 3-manifolds, extending understanding of mapping degrees in 3-manifold topology.

Findings

Finiteness of maps for each nonzero degree

Bounded number of target manifolds for given degree

Results apply to irreducible non-geometric 3-manifolds

Abstract

In this paper, it is shown that every orientable closed 3-manifold maps with nonzero degree onto at most finitely many homeomorphically distinct irreducible non-geometric orientable closed 3-manifolds. Moreover, given any nonzero integer, as a mapping degree up to sign, every orientable closed 3-manifold maps with that degree onto only finitely many homeomorphically distinct orientable closed 3-manifolds.