Rigidity and the lower bound theorem 1

For an arbitrary triangulated (d-1)-manifold without boundary C with f₀ vertices and f₁ edges, define {Mathematical expression}. 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 for d≧4, if γ(C)=0 then C 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 Björner, for the number of k-faces of a triangulated (d-1)-manifold with specified numbers of interior vertices and boundary vertices. (b) If C is a simply connected triangulated d-manifold, d≧4, and γ(lk(v, C))=0 for every vertex v of C, then γ(C)=0. (lk(v,C) is the link of v in C.) (c) Let C be a triangulated d-manifold, d≧3. Then Ske1₁(Δ_d+2) can be embedded in skel₁ (C) iff γ(C)>0. (Δ_d is the d-dimensional simplex.) (d) If P is a 2-simplicial d-polytope then {Mathematical expression}. Related problems concerning pseudomanifolds, manifolds with boundary and polyhedral manifolds are discussed.