Positively curved combinatorial 3-manifolds. (English) Zbl 1201.57013

A combinatorial \(n\)-manifold \(M^{n}\) is a simplicial complex in which the link of each \(k\)-simplex is an \((n-k-1)\)-sphere. The degree of a simplex \(\sigma\in M^{n},\) denoted by \(\deg(\sigma),\) is the number of \(n\)-simplices in \(M^{n}\) having \(\sigma\) as a face. The edge-diameter of \(M^{n},\) written diam\(_{1}(M^{n}),\) is the minimum number of edges needed to connect any vertex in \(M^{n}\) to any other. The edge-distance between vertices \(v,w\in M^{n}\) is the minimum number of edges needed to connect them and will be denoted by \(d_{1}(v,w).\) The first main theorem is a combinatorial version of the classical Bonnet-Myers theorem: Let \(M^{n}\) be a connected, boundaryless, combinatorial \(n\)-manifold in which each \((n-2)\)-simplex has degree at most \(\varepsilon(n)\) where \(\varepsilon(n)\) if \(n=2,3\) and \(\varepsilon(n)=4\) if \(n\geq k.\) Then \(M^{n}\) is compact and has edge-diameter at most \(\delta(n)\) where \(\delta(n)=3\) if \(n=2,\) \(\delta(n)=5\) if \(n=3\) and \(\delta(n)=2\) if \(n\geq4.\) We may also prove that a combinatorial manifold which satisfies the degree bounds of the previous theorem has total angle around each \((n-2)\)-simplex \(\sigma\) less than \(2\pi.\) For this reason such a manifold will be referred to as positively curved. The second main result of the paper is analogous to the rigid sphere theorems of Toponogov and Cheng: Let \(M\) be a positively curved combinatorial \(n\)-manifold. Then the following statements are true: 7mm
If vertices \(v,w\in M\) have edge-distance \(\delta(n)\) then \(M\) is a sphere.
If \(M^{\prime}\) is another such manifold with vertices \(v^{\prime},w^{\prime}\) at edge-distance \(\delta(n)\) and there exists a simplicial isomorphism \(\Psi:Lk(v)\cong Lk(v^{\prime})\) then \(\Psi\) extends to a simplicial isomorphism \(M\cong M^{\prime}\).
For each \((n-1)\)-sphere \(L\) with \((n-3)\)-simplices of degree at most \(\varepsilon(n)\), we explicitly construct a positively curved \(M\) with vertices \(v\) and \(w\) at edge-distance \(\delta(n)\) and \(Lk(v)=L\).


57M99 General low-dimensional topology
53A99 Classical differential geometry
53C99 Global differential geometry
57Q99 PL-topology
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