f-Vectors of 3-Manifolds
Abstract
In 1970, Walkup completely described the set of $f$-vectors for the four 3-manifolds $S^3$, $S^2 twist S^1$, $S^2 \times S^1$, and $RP^3$. 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 $(S^2 twist S^1)^{# k}$, 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2008
- DOI:
- 10.48550/arXiv.0805.1144
- arXiv:
- arXiv:0805.1144
- Bibcode:
- 2008arXiv0805.1144L
- Keywords:
-
- Mathematics - Combinatorics;
- Mathematics - Geometric Topology;
- 57Q15;
- 52B05;
- 57N10;
- 57M50
- E-Print:
- 33 pages, 2 figures, 14 tables, reference updated, to appear in The Electronic Journal of Combinatorics