Rigidity and the lower bound theorem 1

Summary For an arbitrary triangulated (d-1)-manifold without boundaryC withf ₀ vertices andf ₁ edges, define . Barnette proved that (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 ford4, if (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 Björner, 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,d4, and (lk(v, C))=0 for every vertexv ofC, then (C)=0. (lk(v,C) is the link ofv inC.) (c) LetC be a triangulatedd-manifold,d3. Then Ske1₁( _d+2) can be embedded in skel₁ (C) iff (C)>0. ( _d is thed-dimensional simplex.) (d) IfP is a 2-simpliciald-polytope then . Related problems concerning pseudomanifolds, manifolds with boundary and polyhedral manifolds are discussed.