  Motivated by recent developments in quantum simulation of synthetic
dimensions, e.g. in optical lattices of ultracold atoms, we discuss here
$d$-dimensional periodic, gapped quantum systems for $d \le 4$, with focus on
the topology of the occupied energy states. We perform this analysis by asking
whether the spectral subspace below the gap can be spanned by smooth and
periodic Bloch functions, corresponding to localized Wannier functions in
position space. By constructing these Bloch functions inductively in the
dimension, we show that if they are required to be orthonormal then in general
their existence is obstructed by the first two Chern classes of the underlying
Bloch bundle, with the second Chern class characterizing in particular the
4-dimensional situation. If the orthonormality constraint is relaxed, we show
how $m$ occupied energy bands can be spanned by a Parseval frame comprising at
most $m+2$ Bloch functions.
