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Rigidity and the lower bound theorem. I. (English) Zbl 0624.52004
Barnette’s lower bound theorem (LBT) says that for every triangulated \((d- 1)\)-manifold with \(n\) vertices the \(f\)-vector satisfies the inequality \(f_ k\geq \phi_ k(n,d)\) where \(\phi_ k(n,d):=\binom{d}{k}n - \binom{d+1}{k+1}k\) for \(1\leq k\leq d-2\) and \(\phi_{d-1}(n,d):=(d-1)n-(d+1)(d-2).\) It has been conjectured that equality \(f_ k=\phi_ k(n,d)\) for some \(k\) implies that the manifold is a sphere combinatorially equivalent to the boundary complex of a stacked \(d\)-polytope. The main result of the present paper is another proof of the LBT and a proof of this conjecture about the case of equality. As expressed by the title the author uses a rigidity result saying that almost all embeddings of the 1-skeleton of such a manifold into \({\mathbb{R}}^ d\) are rigid (this is called “generically \(d\)-rigid”). The author remarks that the basic relation between the LBT and rigidity has been observed independently by M. Gromov.
Among other results the author discusses manifolds with stacked links, one result saying that every \((d-1)\)-manifold (\(d\geq 5)\) which has only stacked vertex links, must belong to the class \({\mathcal H}^ d\) defined by D. Walkup [Acta Math. 125, 75–107 (1970; Zbl 0204.56301)], in particular it cannot be simply connected. Related results about manifolds with boundary, pseudomanifolds and polyhedral manifolds are also included.
At the end various conjectures are given. One of them relates the number \(\gamma (M):=\min \{f_ 1-\phi_ 1(n,d)\}\) with the first Betti number of \(M\), another one concerns a version of the generalized lower bound conjecture (GLBC) involving the h-vector and the Betti numbers. This remarkable paper is announced to continue with parts II and III.
Reviewer: W.Kühnel

52Bxx Polytopes and polyhedra
57Q15 Triangulating manifolds
05C99 Graph theory
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