We extend several $g$-type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain Kühnel-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 $\hatΔ$ of a manifold with boundary $Δ$; it is obtained from $Δ$ by coning off the boundary of $Δ$ with a single new vertex. We show that despite the fact that $\hatΔ$ 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.