f-Vectors of 3-Manifolds

In 1970, Walkup [46] completely described the set of f -vectors for the four 3manifolds S3, S2×S1, S2×S1, and RP . We improve one of Walkup’s main restricting inequalities on the set of f -vectors of 3-manifolds. As a consequence of a bound by Novik and Swartz [35], we also derive a new lower bound on the number of vertices that are needed for a combinatorial d-manifold in terms of its β1-coefficient, which partially settles a conjecture of Kühnel. Enumerative results and a search for small triangulations with bistellar flips allow us, in combination with the new bounds, to completely determine the set of f -vectors for twenty further 3-manifolds, that is, for the connected sums of sphere bundles (S2×S1)#k and twisted sphere bundles (S2×S1)#k, where k = 2, 3, 4, 5, 6, 7, 8, 10, 11, 14. For many more 3-manifolds of different geometric types we provide small triangulations and a partial description of their set of f -vectors. Moreover, we show that the 3-manifold RP 3#RP 3 has (at least) two different minimal g-vectors. Supported by the DFG Research Group “Polyhedral Surfaces”, Berlin Paritally supported by NSF grant DMS-0600502 the electronic journal of combinatorics 16(2) (2009), #R13 1