On stacked triangulated manifolds

We prove two results on stacked triangulated manifolds in this paper: (a) every stacked triangulation of a connected manifold with or without boundary is obtained from a simplex or the boundary of a simplex by certain combinatorial operations; (b) in dimension $d\ge 4$, if $\mathrm{\Delta }$ is a tight connected closed homology $d$-manifold whose $i$th homology vanishes for $1<i<d-1$, then $\mathrm{\Delta }$ is a stacked triangulation of a manifold.These results give affirmative answers to questions posed by Novik and Swartz and by Effenberger.