Branched coverings of simply connected manifolds

TL;DR

This paper constructs branched double coverings of simply connected manifolds using products of manifolds, providing new insights into their structure and answering a specific topological question up to dimension five.

Contribution

It introduces a method to construct branched double coverings for simply connected manifolds, extending understanding of their topological properties and answering a question of Kotschick and Loeh.

Findings

Every simply connected, closed 4-manifold admits a branched double covering by a circle product and sphere sums.

Every simply connected, closed 5-manifold admits a branched double covering by a circle product and 3-sphere sums.

The degree of the covering map in 5-manifolds relates to the torsion in the second homology group.

Abstract

We construct branched double coverings by certain direct products of manifolds for connected sums of copies of sphere bundles over the 2-sphere. As an application we answer a question of Kotschick and Loeh up to dimension five. More precisely, we show that: (1) every simply connected, closed four-manifold admits a branched double covering by a product of the circle with a connected sum of copies of $S^2 \times S^1$, followed by a collapsing map; (2) every simply connected, closed five-manifold admits a branched double covering by a product of the circle with a connected sum of copies of $S^3 \times S^1$, followed by a map whose degree is determined by the torsion of the second integral homology group of the target.