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Coverings and minimal triangulations of 3-manifolds. (English) Zbl 1229.57010

The authors consider a two-sheeted covering \(\widetilde M\to M\) between two compact, connected and orientable 3-manifolds. Then they compare the complexity of \(\widetilde M\) to that of \(M\). The complexity is defined as the minimum number of tetrahedra in a pseudo-simplicial triangulation. If \(M\) is irreducible with complexity \(c(M)\geq 2\) the main results are the following: The complexities satisfy the inequality \(c(\widetilde M)\leq 2c(M)- 3\) with equality only in the following cases: Either \(\widetilde M= S^3\), \(M=\mathbb{R} P^3\), or \(\widetilde M= L(2k,1)\) for some \(k\geq 2\) and \(M\) is either the lens space \(L(4k,2k- 1)\) or a quotient of \(S^3\) by a generalized quaternion group. Then \(M\) has complexity \(k\), and the minimal triangulations are unique.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
57Q15 Triangulating manifolds
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R05 Triangulating
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References:

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