## 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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