[Paper Review] Real analytic counterexample to the freeness conjecture

Scot Adams|arXiv (Cornell University)|Sep 4, 2015

Mathematical Dynamics and Fractals1 references3 citations

TL;DR

This paper constructs a real analytic counterexample to P. Olver's freeness conjecture for $C^{\ u}$ actions, demonstrating that the action of $\mathbb{Z}$ on a 3-dimensional $C^{\omega}$ manifold fails to be free on any jet bundle of finite order. Using an iterative dynamical surgery process, the author builds a non-simply connected $C^{\omega}$ manifold with a complete $C^{\omega}$ vector field whose $\mathbb{Z}$-action is not free on any jet prolongation, thereby disproving the conjecture in the real analytic category.

ABSTRACT

We provide a counterexample to P.~Olver's freeness conjecture for $C^ω$ transformations.

Motivation & Objective

• To disprove P. Olver's freeness conjecture in the real analytic ($C^{\omega}$) category.
• To construct a $C^{\omega}$ action of $\mathbb{Z}$ on a 3-dimensional manifold that is not free on any finite-order jet bundle.
• To demonstrate that the conjecture fails even for connected Lie groups with noncompact center, via induction from $\mathbb{Z}$-actions.
• To show that the topological complexity of the manifold (infinite fundamental group) prevents freeness in jet prolongations.
• To provide a $C^{\omega}$ counterexample where $C^{\infty}$ counterexamples exist, but with stronger analytic constraints.

Proposed method

• Iteratively construct a nested sequence of subsets $Q_k \subset \mathbb{R}^3$ with increasing topological complexity.
• Define a manifold topology $\tau$ on the union $M = \bigcup Q_k$ by gluing local topologies $\tau_k$ consistently across stages.
• Construct a $C^{\omega}$ atlas $\mathcal{A}$ on $(M, \tau)$ by patching local $C^{\omega}$ charts from each stage.
• Define a complete $C^{\omega}$ vector field $V$ on $M$ that agrees with a constant vector field $E$ on the central region and vanishes at infinity.
• Ensure periodicity of the flow at points $\sigma_k$ to order $k$, using a sequence $\sigma_k$ dense in $M$, to prevent freeness on jet bundles.
• Use a residual set of dense points $\omega_j$ and a minimal index selection rule to ensure $\sigma_k$ remains in open sets, preserving density and analyticity.

Experimental results

Research questions

• RQ1 Does there exist a real analytic ($C^{\omega}$) action of $\mathbb{Z}$ on a manifold that is not free on any finite-order jet bundle?
• RQ2 Can the freeness conjecture fail in the $C^{\omega}$ category, even when it holds in the $C^{\infty}$ category?
• RQ3 Is it possible to construct a $C^{\omega}$ counterexample on a 3-dimensional manifold, rather than higher dimensions?
• RQ4 Can the topological complexity of the manifold (e.g., infinitely generated fundamental group) obstruct freeness in jet prolongations?
• RQ5 Can the rigidity of $C^{\omega}$ structures allow for a counterexample where $C^{\infty}$ counterexamples exist?

Key findings

• A $C^{\omega}$ counterexample to Olver's freeness conjecture exists for the action of $\mathbb{Z}$ on a 3-dimensional $C^{\omega}$ manifold.
• The constructed manifold $M$ has an infinitely generated fundamental group, making it topologically more complex than $\mathbb{R}^4$, and thus not simply connected.
• The $\mathbb{Z}$-action, induced by the flow of a complete $C^{\omega}$ vector field $V$, is not free on any jet bundle of finite order.
• The points $\sigma_k$ are periodic to order $k$, and the set $\Sigma = \{\sigma_1, \sigma_2, \dots\}$ is dense in $M$, ensuring non-freeness at all orders.
• The construction uses a residual set of dense points $\omega_j$ and a minimal index selection to ensure $\sigma_k$ lies in any given open set, preserving analyticity and density.
• The counterexample extends to any connected Lie group with noncompact center via induction from the $\mathbb{Z}$-action, while the conjecture holds for groups with compact center.