Constructions of $d$-spheres from $(d-1)$-spheres and $d$-balls with same set of vertices

Given a combinatorial $(d-1)$-sphere $S$, to construct a combinatorial $d$-sphere $S^{\hspace{.2mm}\prime}$ containing $S$, one usually needs some more vertices. Here we consider the question whether we can do one such construction without the help of any additional vertices. We show that this question has affirmative answer when $S$ is a flag sphere, a stacked sphere or a join of spheres. We also consider the question whether we can construct an $n$-vertex combinatorial $d$-sphere containing a given $n$-vertex combinatorial $d$-ball.

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