f-Vectors of 3-Manifolds

TL;DR

This paper refines the understanding of f-vectors for 3-manifolds, improves bounds, and completely characterizes f-vectors for several classes of 3-manifolds, advancing combinatorial topology knowledge.

Contribution

It improves a key inequality on f-vectors, derives new bounds related to Betti numbers, and completely determines f-vectors for multiple 3-manifold classes.

Findings

Improved inequality on f-vectors of 3-manifolds.

Derived new lower bounds on vertices based on Betti numbers.

Complete characterization of f-vectors for specific 3-manifold classes.

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 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 $(S^2 \times S^1)^{# k}$ 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.