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 $β_2(\mathcal{M})$ of $\mathcal{M}$. Within this framework we conjecture that $v \leq \frac{n}{2}+4$, implying $n \geq 2β_2(\mathcal{M})$. In forthcoming work by the authors, this conjectured bound is shown to be almost tight for all values of $β_2(\mathcal{M})$, 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.