[Paper Review] Computability and the growth rate of symplectic homology

Mark McLean|arXiv (Cornell University)|Sep 21, 2011

Geometric and Algebraic Topology21 references10 citations

TL;DR

This paper demonstrates that there is no algorithm to determine whether certain explicitly constructed Stein manifolds—diffeomorphic to complex affine space of dimension $ n \geq 8 $—are symplectomorphic to one another, by linking symplectic homology growth rates to group-theoretic undecidability. The key result is an undecidability result for symplectic structures on $ \mathbb{R}^{2n} $ and contact structures on $ S^{2n-1} $, arising from the word problem in group theory.

ABSTRACT

For each n greater than 7 we explicitly construct a sequence of Stein manifolds diffeomorphic to complex affine space of dimension n so that there is no algorithm to tell us in general whether a given such Stein manifold is symplectomorphic to the first one or not. We prove a similar undecidability result for contact structures on the 2n - 1 dimensional sphere. We can generalize these results by replacing com- plex affine space with any smooth affine variety of dimension n and the 2n - 1 dimensional sphere with any smooth affine variety intersected with a sufficiently large sphere. We prove these theorems by using an invariant called the growth rate of symplectic homology to reduce these problems to an undecidability result for groups.

Motivation & Objective

• To establish the existence of explicitly constructed Stein manifolds diffeomorphic to $ \mathbb{C}^n $ for $ n \geq 8 $ whose symplectic structures cannot be algorithmically distinguished.
• To extend this undecidability to contact structures on $ S^{2n-1} $, using symplectic completions and Lefschetz fibrations.
• To show that the growth rate of symplectic homology serves as a computable invariant that reduces symplectic classification problems to undecidable group-theoretic problems.
• To prove that no algorithm can determine symplectic equivalence between such manifolds, even when they are diffeomorphic, by encoding the word problem in symplectic topology.
• To generalize the construction to any smooth affine variety of dimension $ n \geq 8 $, and its boundary contact structures on large spheres.

Proposed method

• Constructing finite-type Stein manifolds via explicit Weinstein handle attachments to the standard $ 2n $-dimensional ball, using group presentations as input.
• Using the growth rate of symplectic homology as an invariant to detect non-symplectomorphism, leveraging its algebraic and topological properties.
• Reducing the symplectic classification problem to the undecidability of the word problem in group theory by encoding group presentations into handle attachments.
• Employing Lefschetz fibrations and partial Lefschetz fibrations to control symplectic homology growth and establish upper bounds.
• Applying a maximum principle for holomorphic curves in symplectic cobordisms to rule out non-trivial pseudoholomorphic curves in mapping tori.
• Utilizing exact symplectomorphisms and cylindrical end structures to preserve symplectic homology invariants under deformation.

Experimental results

Research questions

• RQ1 Can an algorithm determine whether two explicitly constructed Stein manifolds, all diffeomorphic to $ \mathbb{R}^{2n} $, are symplectomorphic?
• RQ2 Is the symplectic structure on a Stein manifold determined by its diffeomorphism type when $ n \geq 8 $, given explicit constructions?
• RQ3 Can the word problem in group theory be encoded into symplectic invariants of Stein manifolds via handle attachment?
• RQ4 What role does the growth rate of symplectic homology play in detecting symplectic non-equivalence in high-dimensional Stein manifolds?
• RQ5 Are there undecidability results for contact structures on spheres $ S^{2n-1} $ when the underlying Stein fillings are diffeomorphic to $ \mathbb{R}^{2n} $?

Key findings

• For every group presentation $ P $ with $ n \geq 8 $, there exists a finite-type Stein manifold $ S_P $, explicitly built from $ \mathbb{C}^n $ and Weinstein handles, such that $ S_P \simeq S_{\langle|\rangle} $ if and only if $ G_P $ is trivial.
• The growth rate of symplectic homology is used to distinguish $ S_P $ from $ S_{\langle|\rangle} $, and this invariant is computable in the construction.
• There is no algorithm to determine symplectic equivalence between such $ S_P $, because solving this would solve the word problem in group theory.
• The same undecidability result holds for contact structures on $ S^{2n-1} $, constructed as boundaries of the same Stein manifolds.
• The construction generalizes beyond $ \mathbb{C}^n $: for any smooth affine variety $ V $ of dimension $ n \geq 8 $, similar undecidability holds for $ V $-based Stein manifolds and their boundary contact structures.
• The symplectic homology growth rate is preserved under symplectomorphisms and is sensitive to the fundamental group, which is encoded via handle attachments.