  We study high dimensional expansion beyond simplicial complexes (posets) and
focus on $q$-complexes which are complexes whose basic building blocks are
linear spaces. We show that the complete $q$-complex (consists of all subspaces
of a given linear space) may have non-trivial homology groups and therefore
some techniques for simplicial complexes fail. We develop new techniques to
work bypass this. In particular: (i) We describe a new construction of cones
and use it to determine when the homology of the complete $q$-complex is
trivial. We use this construction to prove the "projective support dimension
conjecture" conjectured by Mnukhin and Siemons. (ii) We define topological high
dimensional expansion for posets, and show that the complete $q$-complex has
linear (in the number of lines) coboundary expansion. (iii) We define the
$q$-Linial-Meshulam model of random $q$-complexes and prove a sharp threshold
for the connectivity of random $q$-complexes.
