Inventiones mathematicae

, Volume 88, Issue 1, pp 125–151

# Rigidity and the lower bound theorem 1

• Gil Kalai
Article

## Summary

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.

## Keywords

Manifold Related Problem Boundary Complex Boundary Vertex Interior Vertex
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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