[Paper Review] Freeness in higher order frame bundles

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

Homotopy and Cohomology in Algebraic Topology2 references3 citations

TL;DR

This paper constructs counterexamples to P. Olver's freeness conjecture for $C^{inity}$ actions on smooth manifolds, demonstrating that for any connected real Lie group with noncompact center, and for the integer group $\mathbb{Z}$, the action fails to be free on any jet bundle of finite order. The key contribution is a $C^\infty$ $G$-action on a manifold $M$ that is fixpoint rare but not free on any finite-order frame bundle, using a limit of vector fields with periodic behavior to all orders.

ABSTRACT

We provide counterexamples to P. Olver's freeness conjecture for $C^\infty$ actions. In fact, we show that a counterexample exists for any connected real Lie group with noncompact center, as well as for the additive group of the integers.

Motivation & Objective

• To disprove P. Olver's freeness conjecture for $C^\infty$ actions on jet bundles of finite order.
• To construct explicit counterexamples for connected real Lie groups with noncompact center and for the additive group $\mathbb{Z}$.
• To demonstrate that fixpoint rare actions may fail to become free on any finite-order frame bundle.
• To analyze the behavior of actions via convergence of vector fields and their flows to all orders at specific points.
• To establish that the conjecture holds only under stronger conditions, such as compact center, and to explore modifications for $C^\omega$ settings.

Proposed method

• Construct a sequence of $C^\infty$ vector fields $V_k$ on $\mathbb{R}^4$ that are complete and periodic to all orders at a dense set of points $\sigma_k$.
• Define a limit vector field $V_\infty$ as the pointwise limit of the $V_k$, ensuring it preserves periodicity to all orders at each $\sigma_k$.
• Use the Baire Category Theorem to show that the set $X = \bigcap_{k=1}^\infty \mathcal{U}(V_k)$ is dense in $\mathbb{R}^4$, where $\mathcal{U}(V_k)$ is the set of points with no nontrivial $V_k$-periodic orbits.
• Induce a $\mathbb{Z}$-action on $\mathbb{R}^4$ via the time-$n$ flow of $V_\infty$, using an isomorphism $f: \mathbb{Z} \to \mathbb{Z}$ to define $z \cdot \sigma = \Phi_{f(z)}^{V_\infty}(\sigma)$.
• Construct a $G$-manifold $M = G \times_Z \mathbb{R}^4$ via induced action, where $G$ is a connected Lie group with noncompact center, ensuring the $G$-action is $C^\infty$ and fixpoint rare.
• Show that the stabilizer of a point in the $k$th-order frame bundle $F_kM$ is infinite by lifting periodicity to all orders and using the structure of the $\mathbb{Z}$-action.

Experimental results

Research questions

• RQ1 Does every $C^\infty$ effective action of a Lie group become free on some finite-order jet bundle?
• RQ2 Can counterexamples to Olver's freeness conjecture be constructed for Lie groups with noncompact center?
• RQ3 Is the $\mathbb{Z}$-action on $\mathbb{R}^4$ via a limit of periodic vector fields free on any finite-order frame bundle?
• RQ4 Can the $C^\infty$ freeness conjecture be modified to hold in the $C^\omega$ category?
• RQ5 What role does the center of a Lie group play in the freeness of its actions on jet bundles?

Key findings

• For any connected real Lie group with noncompact center, there exists a $C^\infty$ action on a manifold that is fixpoint rare but not free on any finite-order frame bundle.
• A counterexample is constructed for the additive group $\mathbb{Z}$, showing the freeness conjecture fails in this case as well.
• The $\mathbb{Z}$-action on $\mathbb{R}^4$ is defined via a limit of vector fields $V_k$ that are periodic to all orders at a dense set of points, ensuring nontrivial stabilizers in frame bundles.
• The stabilizer of a point in the $k$th-order frame bundle $F_kM$ is shown to be infinite, proving the action is not free at any finite order.
• The set $X = \bigcap_{k=1}^\infty \mathcal{U}(V_k)$ is dense in $\mathbb{R}^4$, ensuring the $\mathbb{Z}$-action is fixpoint rare.
• The construction demonstrates that the $C^\infty$ freeness conjecture does not hold in general, even under fixpoint rare conditions, but the $C^\omega$ version holds for groups with compact center.