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The Geometry of Loop Spaces V: Fundamental Groups of Geometric Transformation Groups

Yoshiaki Maeda, Steven Rosenberg

TL;DR

The paper develops a loop-space analytic framework to establish that the fundamental groups of a broad class of geometric transformation groups are infinite. By constructing loop-space forms $\hat{\mathcal{K}}$ from kernels $\hat{k}$ on $M$ and proving a practical criterion (Theorem 2.1) tied to $S^1$-actions, the authors extend prior WCS-based techniques to both finite- and infinite-dimensional groups, including conformal, strict contact, pseudo-Hermitian, and various degree-one symplectic or canonical transformation groups. They demonstrate infinite $\pi_1$ in numerous settings (e.g., $\mathrm{Conf}(S^{4k+1}, g_{\rho})$ for $\rho\neq0$, $\mathrm{Diff}_{\eta,\mathrm{str}}(M)$, $\mathrm{Psh}(M)$, degree-one diffeomorphism groups on $\mathbb{R}^{2k}$ and on cotangent bundles, and Hamiltonian transformation groups on integral symplectic manifolds), revealing a notable discontinuity at special parameter values (e.g., $\rho=0$). The approach provides a versatile analytic tool for probing the topology of transformation groups in CR, contact, and symplectic geometry with potential for further generalizations.

Abstract

We use differential forms on loop spaces to prove that the fundamental group of certain geometric transformation groups is infinite. Examples include both finite and infinite dimensional Lie groups. The finite dimensional examples are the conformal group of $S^{4k+1}$ for a family of nonstandard metrics, and the group of pseudo-Hermitian transformations of a compact CR manifold. Infinite dimensional examples include the group of strict contact diffeomorphisms of a regular contact manifold, and other groups coming from symplectic and contact geometry.

The Geometry of Loop Spaces V: Fundamental Groups of Geometric Transformation Groups

TL;DR

The paper develops a loop-space analytic framework to establish that the fundamental groups of a broad class of geometric transformation groups are infinite. By constructing loop-space forms from kernels on and proving a practical criterion (Theorem 2.1) tied to -actions, the authors extend prior WCS-based techniques to both finite- and infinite-dimensional groups, including conformal, strict contact, pseudo-Hermitian, and various degree-one symplectic or canonical transformation groups. They demonstrate infinite in numerous settings (e.g., for , , , degree-one diffeomorphism groups on and on cotangent bundles, and Hamiltonian transformation groups on integral symplectic manifolds), revealing a notable discontinuity at special parameter values (e.g., ). The approach provides a versatile analytic tool for probing the topology of transformation groups in CR, contact, and symplectic geometry with potential for further generalizations.

Abstract

We use differential forms on loop spaces to prove that the fundamental group of certain geometric transformation groups is infinite. Examples include both finite and infinite dimensional Lie groups. The finite dimensional examples are the conformal group of for a family of nonstandard metrics, and the group of pseudo-Hermitian transformations of a compact CR manifold. Infinite dimensional examples include the group of strict contact diffeomorphisms of a regular contact manifold, and other groups coming from symplectic and contact geometry.

Paper Structure

This paper contains 10 sections, 19 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Differential forms on loop spaces and the fundamental group of $\mathcal{G}(M)$
  3. The Conformal diffeomorphism group of $S^{4k+1}$
  4. Applications to Geometric transformation groups preserving one-forms
  5. The group of strict contact transformations
  6. Generalizations of contactomorphism groups
  7. The group of pseudo-Hermitian transformations
  8. The group of symplectic transformations of homogeneous degree one on $\mathbb{R}^{2k}$
  9. The group of canonical transformations of degree one on the cotangent bundle
  10. Lie group of Hamiltonian symplectic transformations

Key Result

Theorem 1.1

(i) MRT4, MRT2C Let $(M,\omega)$ be an integral symplectic manifold of dimension $4k$, and let $\overline{M}_p$ be the total space of the circle bundle with first Chern class $p\omega$. Then $\overline {M}_p$ admits a Riemannian metric $g_p$ such that for $p\gg 0$, Equivalently, if $\overline{M}$ is a regular contact manifold, then $\overline{M}$ covers infinitely many strictly regular contact ma

Theorems & Definitions (30)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.4
  • ...and 20 more