# Four-manifolds with shadow-complexity one

## TL;DR

This paper classifies all closed oriented smooth 4-manifolds with a specific shadow-complexity of at most one, revealing their structure through basic building blocks and their doubles, and contrasting with the zero complexity case.

## Contribution

It characterizes all 4-manifolds with shadow-complexity one using a finite set of basic blocks and explores their properties and representations.

## Key findings

- All such 4-manifolds are generated by 20 basic blocks.
- Many are doubles of 2-handlebodies and can be encoded via finite group presentations.
- Complexity one includes doubles not arising from 2-handlebodies, like RP^3 imes S^1.

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