Twisted face-pairing 3-manifolds (Q2781411)

The space obtained by a face-pairing of a faceted 3-ball \(P\) (that is by indentifying in pairs the faces of a polyhedron) may not be a 3-manifold in some of the vertices. In a previous paper [Math. Res. Lett. 7, No. 4, 477-491 (2000; Zbl 0958.57021)], the authors considered a notion of twisted face pairing, by subdividing the edges of \(P\) and composing the identifying homeomorphisms with twist maps (rotations) of the subdivided faces (respecting orientations), then showing by an Euler-characteristic argument that now the result is always a closed 3-manifold (so the method yields infinite parametrized families of closed 3-manifolds). In the present paper, which is an enriched version of the previous introductory paper, a geometric and more conceptual proof of this result is given by showing that the quotient complex has just one vertex whose link is isomorphic with the faceted sphere dual to the subdivided polyhedron. Also it is shown that, by simultaneously reversing the direction of all twists in the face identifications, one obtains the same 3-manifold, and more specifically dual cell structures of the same manifold. Further, it is shown that the fundamental group of such a twisted face pairing 3-manifold surjects onto the original pseudomanifold obtained from \(P\), with deleted vertices, and the homology groups of these spaces are compared. Various examples of twisted face-pairing manifolds are given, including manifolds from five of Thurston's eight 3-dimensional geometries. Two further preprints of the authors' are cited relating the construction to Heegaard diagrams, surgery on links and Gromov hyperbolicity.