Four-manifolds with shadow-complexity one

Koda, Yuya; Martelli, Bruno; Naoe, Hironobu

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 31 (2022) no. 4, pp. 1111-1212.

Nous étudions expérimentalement l’ensemble de toutes les 4-variétés lisses orientées fermées, selon une complexité définie à l’aide des ombres de Turaev. Cette complexité mesure à peu près à quel point le 2-squelette de la 4-variété est compliqué.

Nous caractérisons ici toutes les 4-variétés orientées fermées qui ont complexité mineure ou égale à 1. Ces variétés sont engendrées par un certain ensemble de 20 blocs, qui sont des 4-variétés relativement simples avec un bord constitué de copies de $S^2\times S^1$, plus des sommes connexes avec des copies de ${\mathrm{ℂ\mathbb{P}}}^2$ avec orientation arbitraire.

Toutes les variétés générées par ces blocs sont des doubles. Beaucoup d’entre elles sont des doubles de corps à 2-anses et sont donc efficacement codifiées en utilisant des présentations finies de groupes. Contrairement au cas de complexité zéro, en complexité 1 il y a aussi beaucoup de doubles qui ne sont pas doubles de corps à 2-anses, comme par exemple ${\mathrm{ℝ\mathbb{P}}}^3\times S^1$.

We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev’s shadows. This complexity roughly measures how complicated the 2-skeleton of the 4-manifold is.

We characterise here all the closed oriented 4-manifolds that have complexity at most one. They are generated by a certain set of 20 blocks, that are some basic 4-manifolds with boundary consisting of copies of $S^2\times S^1$, plus connected sums with some copies of ${\mathrm{ℂ\mathbb{P}}}^2$ with either orientation.

All the manifolds generated by these blocks are doubles. Many of these are doubles of 2-handlebodies and are hence efficiently encoded using finite presentations of groups. In contrast to the complexity zero case, in complexity one there are also plenty of doubles that are not doubles of 2-handlebodies, like for instance ${\mathrm{ℝ\mathbb{P}}}^3\times S^1$.

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