Inventiones mathematicae

, Volume 88, Issue 1, pp 125–151 | Cite as

Rigidity and the lower bound theorem 1

  • Gil Kalai


For an arbitrary triangulated (d-1)-manifold without boundaryC withf0 vertices andf1 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 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,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(Δd+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.


Manifold Related Problem Boundary Complex Boundary Vertex Interior Vertex 
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Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • Gil Kalai
    • 1
    • 2
  1. 1.Institute of MathematicsHebrew UniversityJerusalemIsrael
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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