  We extend several $g$-type theorems for connected, orientable homology
manifolds without boundary to manifolds with boundary. As applications of these
results we obtain K\"uhnel-type bounds on the Betti numbers as well as on
certain weighted sums of Betti numbers of manifolds with boundary. Our main
tool is the completion $\hat\Delta$ of a manifold with boundary $\Delta$; it is
obtained from $\Delta$ by coning off the boundary of $\Delta$ with a single new
vertex. We show that despite the fact that $\hat{\Delta}$ has a singular
vertex, its Stanley--Reisner ring shares a few properties with the
Stanley--Reisner rings of homology spheres. We close with a discussion of a
connection between three lower bound theorems for manifolds, PL-handle
decompositions, and surgery.
