[Paper Review] Four-manifolds with shadow-complexity one

Yuya Koda, Bruno Martelli+1 more|arXiv (Cornell University)|Mar 18, 2018

Geometric and Algebraic Topology21 references3 citations

TL;DR

This paper characterizes all closed, oriented, smooth 4-manifolds with shadow-complexity at most one using Turaev's shadow theory. It shows these manifolds are generated by 20 specific blocks with boundary $S^2 \times S^1$ and connected sums with $\mathbb{CP}^2$ (in either orientation), and includes both doubles of 2-handlebodies and non-2-handlebody doubles like $\mathbb{RP}^3 \times S^1$, providing a finite, systematic classification at low complexity.

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 $S^2 imes S^1$, 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 $\mathbb{RP}^3 imes S^1$.

Motivation & Objective

• To develop an experimental, combinatorial classification of closed oriented smooth 4-manifolds using Turaev's shadow complexity.
• To identify and characterize all 4-manifolds with connected shadow complexity at most one.
• To determine whether such manifolds are doubles of 2-handlebodies or include more general doubles.
• To establish a finite generating set of basic 4-manifolds with $S^2 \times S^1$ boundary that produce all low-complexity 4-manifolds.

Proposed method

• Uses Turaev's shadow theory to define a complexity invariant $c^*(M)$ measuring the complexity of the 2-skeleton of a 4-manifold.
• Applies a systematic simplification procedure to shadow surfaces via moves on sequences of integers and flat vertices.
• Employs a finite set of 20 basic 4-manifolds with boundary $S^2 \times S^1$ as building blocks.
• Uses connected sums with $\mathbb{CP}^2$ (in either orientation) to generate all 4-manifolds of complexity ≤1.
• Applies handlebody theory and group presentations to encode doubles efficiently, especially those of 2-handlebodies.
• Verifies that all configurations reduce to known blocks via explicit moves, including those involving $-2$-curves and flat vertices.

Experimental results

Research questions

• RQ1 Which closed oriented smooth 4-manifolds have shadow-complexity at most one?
• RQ2 Can all such 4-manifolds be generated from a finite set of basic blocks with $S^2 \times S^1$ boundary?
• RQ3 Are all 4-manifolds of complexity ≤1 doubles of 2-handlebodies, or do some arise as doubles of more general 4-manifolds?
• RQ4 What role does $\mathbb{CP}^2$ (with either orientation) play in generating low-complexity 4-manifolds?
• RQ5 How do shadow moves and simplification rules classify the structure of these 4-manifolds?

Key findings

• All closed oriented smooth 4-manifolds with shadow-complexity at most one are generated by 20 specific blocks with boundary $S^2 \times S^1$ and connected sums with $\mathbb{CP}^2$ in either orientation.
• The set includes both doubles of 2-handlebodies and non-2-handlebody doubles, such as $\mathbb{RP}^3 \times S^1$, which is not a double of a 2-handlebody.
• The complexity $c^*(M)$ is finite and low, with $c^*(M) = 0$ or $1$ for many manifolds, including $S^4$, $S^2 \times S^2$, and $\mathbb{CP}^2$, all of which have $c^* = 0$.
• The construction via group presentations and thickenings yields finite sets of 4-manifolds with identical $\pi_1$, $\pi_2$, and homology, and these are precisely the doubles of 4-dimensional thickenings.
• The classification is complete and finite: only finitely many 4-manifolds exist with $c^*(M) \leq 1$, and they are all generated from the 20 blocks.
• Simplification moves on shadow sequences (e.g., via Figures 46, 50, 51) reduce all configurations to the generating blocks, proving completeness of the classification.