Branched coverings of simply connected manifolds

Abstract

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 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 × 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 × S 1 , followed by a map whose degree is determined by the torsion of the second integral homology group of the target.

Ordering Thurston’s geometries by maps of nonzero degree

“…Similarly, the non-aspherical geometries are not included in the above theorem. Those geometries are either products or their representatives are simply connected; see [15,17] for a discussion.…”

The branched deformations of the special Lagrangian submanifolds

“…We define V = χ V and write µ s : U L → U L be the 1-parameter family of diffeomorphisms such that dµs ds = V • µ s with µ := µ s | s=1 . We would like to consider the special Lagrangian equation over the Calabi-Yau structure (U L , µ ⋆ J, µ ⋆ ω, µ ⋆ Ω) for a nondegenerate pair a := α + + α − ⊂ U ′ L , which is defined to be (39) SL λ ⋆ g (a) := ⋆ λ ⋆ g ι ⋆ a (µ ⋆ ℑΩ). Similarly, we could define…”

“…, where SL g k,t (A k ) is the special Lagrangian equation for the variation of Calabi-Yau structure ϕ ⋆ k,t (U L , J, ω, Ω) defined in (39) with Calabi-Yau metric ϕ ⋆ k,t g U L and g k,t := ϕ ⋆ k,t g is the induced Riemannian metric on the zero section. As α + 1 is a harmonic 1-form, we could also write α + 1 = df + 1 , where f + 1 is a section of a affine line bundle.…”

Fundamental groups of aspherical manifolds and maps of non-zero degree

“…We note that we could have included non-aspherical geometries as well in the above statement, however those geometries are not interesting for the domination-by-products question, either because they are products themselves or because their representatives are simply connected. The latter geometries were contained as trivial examples in [26], where we constructed maps from products to every simply connected 4-manifold.…”

“…Here, we deal only with the aspherical geometries, because the non-aspherical ones are not interesting for domination by products. Namely, the non-aspherical geometries are either products of a sphere with a non-compact factor (H 2 × S 2 , R 2 × S 2 , S 3 × R), or compact themselves (S 2 × S 2 , CP 2 , S 4 ), and all of their representatives are dominated by products [17,20,26].…”