[Paper Review] Regularity and nearness theorems for families of local Lie groups

Tom McGaffey|arXiv (Cornell University)|Jun 14, 2012

Mathematical and Theoretical Analysis19 references3 citations

TL;DR

This paper establishes nonstandard analysis-based regularity and nearness theorems for families of local Lie groups, proving that pointwise convergent, differentiable families converge smoothly—up to coordinate change—to an analytic local Lie group. The key contribution is a nonstandard regularity result showing that S-continuous Lie group structures imply analyticity after coordinate transformation, advancing the local version of Hilbert’s Fifth Problem via nonstandard methods and a novel topology on map germs.

ABSTRACT

In this work, we prove three types of results with the strategy that, together, the author believes these should imply the local version of Hilbert's Fifth problem. In a separate development, we construct a nontrivial topology for rings of map germs on Euclidean spaces. First, we develop a framework for the theory of (local) nonstandard Lie groups and within that framework prove a nonstandard result that implies that a family of local Lie groups that converge in a pointwise sense must then differentiability converge, up to coordinate change, to an analytic local Lie group, see corollary 6.1. The second result essentially says that a pair of mappings that almost satisfy the properties defining a local Lie group must have a local Lie group nearby, see proposition 7.1. Pairing the above two results, we get the principal standard consequence of the above work, corollary 7.2, which can be roughly described as follows. If we have pointwise equicontinuous family of mapping pairs (potential local Euclidean topological group structures), pointwise approximating a (possibly differentiably unbounded) family of differentiable (suffi- ciently approximate) almost groups, then the original family has, after appropriate coordinate change, a local Lie group as a limit point. The third set of results give nonstandard renditions of equicontinuity criteria for families of differentiable functions, see theorem 9.1. These results are critical in the proofs of the principal results of this thesis as well as the standard interpretations of the main results here. Following this material, we have a long chapter constructing a Hausdorff topology on the ring of real valued map germs on Euclidean space. This topology has good properties with respect to convergence and composition. See the detailed introduction to this chapter for the motivation and description of this topology.

Motivation & Objective

• To establish a nonstandard framework for local Lie groups to prove regularity and nearness theorems.
• To resolve the local version of Hilbert’s Fifth Problem by showing that pointwise equicontinuous families of almost local Lie groups converge to analytic local Lie groups after coordinate change.
• To construct a Hausdorff topology on rings of real-valued map germs on Euclidean spaces with good convergence and composition properties.
• To provide nonstandard equivalents of equicontinuity criteria for differentiable functions, essential for proving the main results.
• To demonstrate that S-continuity of the adjoint map implies analyticity of the standard part of a local *Lie group.

Proposed method

• Utilizes nonstandard analysis (NSA) with transfer, saturation, and overflow principles to analyze local *Lie groups and their standard parts.
• Applies S-regularity and S-analyticity concepts to show that S-continuous group operations imply analyticity after coordinate change.
• Employs the *exponential map and its regularity to prove that the standard part of a *Lie group is a local Lie group.
• Introduces a new Hausdorff topology on the ring of real-valued map germs on Euclidean spaces, ensuring convergence and composition compatibility.
• Uses the *Jordan separation theorem and transfer of topological properties to bound image sizes of germs and prove inclusion results.
• Applies the *μ-exp lemma and differential geometry of *GL(n) to establish S-continuity of the adjoint map, a key step in proving analyticity.

Experimental results

Research questions

• RQ1 Can a family of local Lie groups that converges pointwise in a differentiable sense be shown to converge smoothly to an analytic local Lie group?
• RQ2 If two mappings almost satisfy the axioms of a local Lie group, does there exist a true local Lie group nearby?
• RQ3 Can nonstandard equicontinuity criteria be formulated and used to prove regularity theorems for families of differentiable functions?
• RQ4 Does a Hausdorff topology exist on the ring of map germs on Euclidean spaces that respects convergence and composition?
• RQ5 Can the standard part of a *Lie group structure be shown to be analytic under S-continuity and appropriate coordinate changes?

Key findings

• A family of local Lie groups that is pointwise equicontinuous and approximates differentiable almost groups converges, after coordinate change, to an analytic local Lie group (Corollary 7.2).
• The standard part of a σ-local *Lie group satisfying the SC∘ condition is an analytic local Lie group in some coordinate system (Theorem 6.2).
• The adjoint map on a *Lie group is S-continuous, which implies that the standard part of the group is analytic (Theorem 6.2).
• A nontrivial Hausdorff topology is constructed on the ring of real-valued map germs on ℝⁿ, with good convergence and composition properties (Chapter 10).
• Germs of homeomorphisms act as homeomorphisms on germ spaces, and their images uniformly contain small balls, independent of the germ (Proposition 10.15).
• The *exponential map is regular, and its differential (the adjoint map) is S-continuous, which is critical for proving analyticity of the standard part (Section 4).