Neighborly combinatorial 3-manifolds with dihedral automorphism group

It is well known that for anyn≧5 the boundary complex of the cyclic 4-polytopeC(n, 4) is a neighborly combinatorial 3-sphere admitting a vertex transitive action of the dihedral groupD _n of order 2n. In this paper we present a similar series of neighborly combinatorial 3-manifolds withn≧9 vertices, each homeomorphic to the “3-dimensional Klein bottle”. Forn=9 andn=10 these examples have been observed. before by A. Altshuler and L. Steinberg. Moreover we give a computer-aided enumeration of all neighborly combinatorial 3-manifolds with such a symmetry and with at most 19 vertices. It turns out that there are only four other types, one with 10, 15, 17, 19 vertices. We also discuss the more general case of manifolds with cyclic automorphism groupC _n.