f-Vectors of 3-Manifolds

In 1970, Walkup completely described the set of $f$-vectors for the four 3-manifolds $S^3$, $S^{2}twist{S}^1$, $S^2\times S^1$, and $R{P}^3$. 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, we also derive a new lower bound on the number of vertices that are needed for a combinatorial $d$-manifold in terms of its ${\beta }_1$-coefficient, which partially settles a conjecture of K"uhnel. 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 $(S^2 \times S^1)^{# k}$andtwistedspherebundles$(S^2 twist S^1)^{# k}$,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.