[Paper Review] Diffeomorphism groups of balls and spheres

Kathryn Mann|arXiv (Cornell University)|Apr 9, 2013

Geometric and Algebraic Topology8 references3 citations

TL;DR

This paper investigates the algebraic and dynamical structure of diffeomorphism groups of spheres and balls, proving that the identity component of smooth diffeomorphisms of odd-dimensional spheres admits no nontrivial homomorphisms to the group of $C^1$ diffeomorphisms of any ball. It generalizes results of Ghys and Herman, constructs a finitely generated, torsion-free group acting smoothly on $S^1$ that does not extend to a $C^1$ action on the 2-ball, and establishes a fundamental algebraic obstruction to extending group actions from spheres to balls.

ABSTRACT

In this paper we discuss the relationship between groups of diffeomorphisms of spheres and balls. We survey results of a topological nature and then address the relationship as abstract (discrete) groups. We prove that the identity component Diff_0(S^{2n-1}) of the group of smooth diffeomorphisms of S^{2n+1} admits no nontrivial homomorphisms to the group of C^1 diffeomorphisms of the ball B^m for any n and m. This result generalizes theorems of Ghys and Herman. We also examine finitely generated subgroups of Diff_0(S^n) and produce an example of a finitely generated torsion free group Gamma with an action on the circle by smooth diffeomorphisms that does not extend to a C^1 action of Gamma on the disc.

Motivation & Objective

• To understand the algebraic and dynamical differences between diffeomorphism groups of spheres and balls.
• To investigate whether actions of finitely generated groups on spheres can be extended to actions on the ball via $C^1$ diffeomorphisms.
• To resolve the existence of nontrivial group homomorphisms from $\operatorname{Diff}^\infty_0(S^{2n-1})$ to $\operatorname{Diff}^1_0(B^m)$.
• To construct explicit examples of smooth group actions on $S^1$ that do not extend to $C^1$ actions on $B^2$.
• To generalize and extend results of Ghys and Herman on the nonexistence of group-theoretic sections for diffeomorphism groups.

Proposed method

• Use finite-order diffeomorphisms on odd-dimensional spheres to construct subgroups with non-extendable actions.
• Apply representation theory and matrix centralizer arguments to show that certain groups cannot admit faithful linear representations.
• Employ inductive reduction on fixed-point sets of iterated $p$-power diffeomorphisms to derive contradictions.
• Leverage results from Franks and Handel on distortion elements in finite groups to construct the key example.
• Analyze the structure of normal subgroups in diffeomorphism groups to show that any homomorphism must be trivial.
• Use the cone construction and isotopy extension to compare topological vs. smooth extension properties.

Experimental results

Research questions

• RQ1 Does there exist a nontrivial group homomorphism from $\operatorname{Diff}^\infty_0(S^{2n-1})$ to $\operatorname{Diff}^1_0(B^m)$ for any $n$ and $m$?
• RQ2 Can a finitely generated, torsion-free group acting smoothly on $S^1$ always be extended to a $C^1$ action on $B^2$?
• RQ3 What is the role of finite-order diffeomorphisms in obstructing the extension of group actions from spheres to balls?
• RQ4 Are there algebraic obstructions that prevent the existence of group-theoretic sections for the restriction map $\pi: \operatorname{Diff}^r_0(B^{n+1}) \to \operatorname{Diff}^r_0(S^n)$?
• RQ5 How do the normal subgroup structures of diffeomorphism groups on spheres and balls differ in the smooth category?

Key findings

• There is no nontrivial group homomorphism from $\operatorname{Diff}^\infty_0(S^{2k-1})$ to $\operatorname{Diff}^1_0(B^m)$ for any $k \geq 1$ and $m$, generalizing a result of Herman.
• A finitely generated, torsion-free group $\Gamma$ exists that acts smoothly on $S^1$ but does not extend to a $C^1$ action on $B^2$, demonstrating a sharp contrast with $C^\infty$ extensions.
• The group $\operatorname{Diff}^\infty_0(S^{2n-1})$ has no nontrivial linear representations into $\operatorname{GL}(m,\mathbb{C})$ that factor through the action on the ball, due to centralizer and normal subgroup obstructions.
• The only normal subgroups of the derived subgroup of a certain finite-order diffeomorphism group on $S^{2n-1}$ are the cyclic groups generated by powers of the generator, which leads to a contradiction under linearization.
• The construction of the non-extendable action relies on the existence of free, finite-order diffeomorphisms on odd-dimensional spheres and their lifting properties.
• The failure of extension is fundamentally algebraic and not topological, as topological sections (e.g., via cone maps) do exist in the homeomorphism category.