$f$-Vectors of $3$-Manifolds

Abstract

In 1970, Walkup completely described the set of $f$-vectors for the four $3$-manifolds $S^3$, $S^2\rlap{\times}\_\;S^1$, $S^2\!\times\!S^1$, and ${\Bbb R}{\bf P}^{\,3}$. 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 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 $(S^2\!\times\!S^1)^{\# k}$ and twisted sphere bundles $(S^2\rlap{\times}\_\;S^1)^{\# k}$, 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 ${\Bbb R}{\bf P}^{\,3}\#\,{\Bbb R}{\bf P}^{\,3}$ has (at least) two different minimal $g$-vectors.