  In this paper we discuss face numbers of generalised triangulations of
manifolds in arbitrary dimensions. This is motivated by the study of
triangulations of simply connected $4$-manifolds: We observe that, for a
triangulation $\mathcal{T}$ of a simply connected $4$-manifold $\mathcal{M}$
with $n$ pentachora, an upper bound on the number of vertices $v$ of
$\mathcal{T}$ as a function of $n$ yields a lower bound for $n$ depending only
on the second Betti number $\beta_2(\mathcal{M})$ of $\mathcal{M}$. Within this
framework we conjecture that $v \leq \frac{n}{2}+4$, implying $n \geq
2\beta_2(\mathcal{M})$. In forthcoming work by the authors, this conjectured
bound is shown to be almost tight for all values of $\beta_2(\mathcal{M})$,
with a gap of at most two.
  We extend our conjecture to arbitrary dimensions and show that an $n$-facet
triangulation of an odd-dimensional $d$-manifold, $n \geq d$, can have at most
$n + \frac{d-1}{2}$ vertices, and conjecture that, for $d$ even, the bound is
$\frac{n}{2}+d$. We show that these (conjectured) bounds are (would be) tight
for all odd (even) dimensions and all values of $n \geq d$. Finally, we give
necessary conditions for the dual graph of $\mathcal{T}$ to satisfy our
conjecture. We furthermore present families of $4$-dimensional pseudomanifolds
with singularities in their edge links that have more than $\frac{n}{2}+4$
vertices, thereby proving that the manifold condition is necessary for our
conjecture to hold.
