  We prove a number of new restrictions on the enumerative properties of
homology manifolds and semi-Eulerian complexes and posets. These include a
determination of the affine span of the fine $h$-vector of balanced
semi-Eulerian complexes and the toric $h$-vector of semi-Eulerian posets.
  The lower bounds on simplicial homology manifolds, when combined with higher
dimensional analogues of Walkup's 3-dimensional constructions \cite{Wal}, allow
us to give a complete characterization of the $f$-vectors of arbitrary
simplicial triangulations of $S^1 \times S^3, \C P^2,$ $ K3$ surfaces, and
$(S^2 \times S^2) # (S^2 \times S^2).$ We also establish a principle which
leads to a conjecture for homology manifolds which is almost logically
equivalent to the $g$-conjecture for homology spheres. Lastly, we show that
with sufficiently many vertices, every triangulable homology manifold without
boundary of dimension three or greater can be triangulated in a 2-neighborly
fashion.
