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**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.

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.

Reviewer: Darryl McCullough (Norman)

### 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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\textit{B. Martelli} and \textit{C. Petronio}, Exp. Math. 10, No. 2, 207--236 (2001; Zbl 1050.57018)

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