## 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
(1)
If vertices $$v,w\in M$$ have edge-distance $$\delta(n)$$ then $$M$$ is a sphere.
(2)
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 $$M\cong M^{\prime}$$.
(3)
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 $$Lk(v)=L$$.

### MSC:

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