For an arbitrary triangulated (d-1)-manifold without boundaryC withf 0 vertices andf 1 edges, define $$\gamma (C) = f_1 - df_0 + \left( {\begin{array}{*{20}c} {d + 1} \\ 2 \\ \end{array} } \right)$$ . Barnette proved that γ(C)≧0. We use the rigidity theory of frameworks and, in particular, results related to Cauchy's rigidity theorem for polytopes, to give another proof for this result. We prove that ford≧4, if γ(C)=0 thenC is a triangulated sphere and is isomorphic to the boundary complex of a stacked polytope. Other results: (a) We prove a lower bound, conjectured by Bjorner, for the number ofk-faces of a triangulated (d-1)-manifold with specified numbers of interior vertices and boundary vertices. (b) IfC is a simply connected triangulatedd-manifold,d≧4, and γ(lk(v, C))=0 for every vertexv ofC, then γ(C)=0. (lk(v,C) is the link ofv inC.) (c) LetC be a triangulatedd-manifold,d≧3. Then Ske11(Δ d+2) can be embedded in skel1 (C) iff γ(C)>0. (Δ d is thed-dimensional simplex.) (d) IfP is a 2-simpliciald-polytope then $$f_1 (P) \geqq df_0 (P) - \left( {\begin{array}{*{20}c} {d + 1} \\ 2 \\ \end{array} } \right)$$ . Related problems concerning pseudomanifolds, manifolds with boundary and polyhedral manifolds are discussed.