The structure of closed nonpositively curved euclidean cone 3-manifolds (Q1322104)

A metric space is called a Euclidean cone 3-manifold if it is obtained as quotient space of a disjoint union of geodesic 3-simplices in \({\mathbf E}^ 3\) by an isometric pairing of dimension-two faces that gives a manifold as underlying topological space. The cone locus of such a manifold is a set of points where the sum of dihedral angles around the point is not \(2 \pi\). There is a flat Riemannian metric on the union of the top-dimensional cells of the simplices and the codimension-1 cells which may be extended smoothly over those 0-cells whose angle is \(2 \pi\). If the cone angles are all greater than \(2 \pi\) the author shows that a cone manifold possesses a smooth Riemannian metric of nonpositive sectional curvature. Let \(M\) be a closed orientable Euclidean cone 3-manifold with cone locus a link (no vertices) all of whose cone angles \(>2 \pi\). The author proves that \(M\) contains a canonical 2-complex \(C\) such that all components of \(\overline {M \backslash C}\) are Seifert-fibered 3-manifolds each possessing a convex Euclidean cone metric. Moreover, \(M\) is itself atoroidal iff each of these components is a solid torus. The result gives a decomposition of an Euclidean cone 3-manifold similar to that of Jaco- Shalen-Johannson decomposition of a 3-manifold. Numerous examples of Euclidean cone manifolds one can get as branched covers over Euclidean orbifolds, in particular some of them are given in the paper which are branched coverings over \(S^ 3\) branched over the figure-eight knot with branching indices greater than 2.