## Three-manifolds having complexity at most 9.(English)Zbl 1050.57018

This paper concerns a complexity function $$c(M)$$ for orientable 3-manifolds, that was developed and initially studied by S. V. Matveev. It is defined to be the minimal number of vertices of a “simple” spine of $$M$$. For closed $$\mathbb{P}^2$$-irreducible $$3$$-manifolds (other than $$S^3$$, $$\mathbb{RP}^3$$, and $$L(3,1)$$), $$c(M)$$ is the minimal number of tetrahedra in a nonregular triangulation of $$M$$, so there are only finitely many such $$3$$-manifolds of a given complexity.
This paper explains the authors’ algorithm to list the closed orientable irreducible $$3$$-manifolds of complexity up to $$9$$, and gives the results of its implementation. In particular, they find $$1156$$ manifolds of complexity $$9$$: $$272$$ lens spaces, $$873$$ graph manifolds, $$7$$ torus bundles over $$S^1$$, and $$4$$ hyperbolic manifolds. The latter are the four closed hyperbolic $$3$$-manifolds of smallest known volume. In more recent work [Complexity of geometric three-manifolds, Mathematics ArXiv GT/0303249, to appear in Geom. Dedicata], the authors have continued the development of these ideas. They also have corrected a slight error in the paper under review: in [B. Martelli, Complexity of $$3$$-manifolds, Mathematics ArXiv GT/0405250] it is noted that two graph manifolds of complexity $$9$$ appeared twice in the list of $$1156$$, so the actual count is $$1154$$.
The authors’ approach is to decompose the $$3$$-manifolds into simpler pieces. They extend the definition of complexity to manifolds that are marked by a specific choice (up to isotopy) of an essentially imbedded theta-graph in each torus boundary component. Gluing two marked manifolds together by identifying one pair of torus boundary components respecting the markings is called an assembling. Also, there are self-assemblings that identify two torus boundary components of the same marked 3-manifold. Since the spines are compatible with the markings, they fit together to give a spine for the assembled manifold. In the case of a self-assembling, six new vertices must be added in constructing the new spine.
An assembling is called sharp if the complexities add, and there is a similar notion for self-assemblings. A 3-manifold is called a “brick” if it does not result from any sharp assembling or self-assembling, other than the trivial assembling with an $$S^1\times S^1\times I$$ having parallel theta graphs in its two boundary tori. Every prime, marked $$3$$-manifold is obtained by sharp assemblings and self-assemblings of bricks.
The main work of the paper is to find the $$30$$ bricks ($$19$$ of these are closed, so are irrelevant for assembling) of complexity up to $$9$$. Some parts of the determination are computer-assisted, but most of the effort is a very careful analysis of the behavior of spines under the different possible gluings of the low-complexity bricks.

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M99 General low-dimensional topology 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds

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