Four-manifolds with shadow-complexity one

Abstract

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 ${\mathbb{CP}}^2$ 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 ${\mathbb{RP}}^3\times S^1$.

arXiv comment75 pages, 77 figures; introduction expanded in the second version

References in corpus (7)

• Genus two trisections are standard
• Trisections and spun 4-manifolds
• Mazur manifolds and corks with small shadow complexities
• Topological 4-manifolds with geometrically 2-dimensional fundamental groups
• Corks with large shadow-complexity and exotic 4-manifolds
• The number of smooth 4-manifolds with a fixed complexity
• Complexity of PL-manifolds

Cited by in corpus (3)

• Shadows of acyclic 4-manifolds with sphere boundary
• A new explicit way of obtaining special generic maps into the 3-dimensional Euclidean space
• Milnor fibration, A'Campo's divide and Turaev's shadow