$g$-vectors of manifolds with boundary

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.