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{\bf Frank H. Lutz, Thom Sulanke and Ed Swartz}
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{\bf $f$-Vectors of $3$-Manifolds}
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In 1970, Walkup completely described the set of $f$-vectors for the
four $3$-manifolds $S^3$,
$S^2\hbox{$\times\kern-1.62ex\_\kern-.4ex\_\kern.7ex$}S^1$,
$S^2\!\times\!S^1$, 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 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
$(S^2\hbox{$\times\kern-1.62ex\_\kern-.4ex\_\kern.7ex$}S^1)^{\#
k}$, 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.

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