A splitting theorem for good complexifications

TL;DR

This paper establishes a splitting theorem for manifolds with good complexifications, revealing their structure as fiber bundles over tori with fibers that also admit good complexifications and have zero virtual first Betti number.

Contribution

It introduces a Cheeger-Gromoll type splitting theorem for manifolds with good complexifications, providing new restrictions on their fundamental groups and structure.

Findings

Manifolds with good complexifications have finite covers that split as fiber bundles over tori.

Fibers in these bundles admit good complexifications and have zero virtual first Betti number.

Applications are provided for manifolds of dimension up to 5.

Abstract

The purpose of this paper is to produce restrictions on fundamental groups of manifolds admitting good complexifications by proving the following Cheeger-Gromoll type splitting theorem: Any closed manifold $M$ admitting a good complexification has a finite-sheeted regular covering $M_1$ such that $M_1$ admits a fiber bundle structure with base $(S^1)^k$ and fiber $N$ that admits a good complexification and also has zero virtual first Betti number. We give several applications to manifolds of dimension at most 5.