Branched coverings of simply connected manifolds.(English)Zbl 1302.57002

Summary: 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 Löh 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.

MSC:

 57M05 Fundamental group, presentations, free differential calculus 57M12 Low-dimensional topology of special (e.g., branched) coverings 55M25 Degree, winding number
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References:

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