The Geometry of Loop Spaces III: Isometry Groups of Contact Manifolds

TL;DR

This work identifies infinite fundamental groups of isometry groups for a family of high-dimensional circular contact manifolds $ar M_p$ by exploiting Wodzicki-Chern-Simons forms on loop spaces. By computing the curvature of the circle bundles and showing the top WCS term on the loop space is a nonzero polynomial in $p^2$ with nonvanishing leading coefficient, the authors prove $ ext{π}_1( ext{Isom}(ar M_p))$ is infinite for large $|p|$. The Kodaira-Thurston example provides a concrete non-Kähler setting where the top WCS class is nonzero for all $p$, while the Kähler case reveals nonvanishing Wodzicki-Pontryagin forms in dimensions $4k+2$ on loop spaces, including lens spaces, highlighting new infinite-dimensional characteristic classes. Together, these results extend known circle-bundle isometry phenomena to arbitrarily high dimensions and uncover nontrivial Wodzicki invariants arising from loop-space geometry. The paper blends explicit curvature computations, computer verifications, and high-dimensional geometric constructions to establish a robust link between fiberwise circle actions, loop-space invariants, and isometry group topology.

Abstract

We study the isometry groups of manifolds $\overline {M}_p$, $p\in\mathbb{Z}$, which are closed contact $(4n+1)$-manifolds with closed Reeb orbits. Equivalently, $\overline{M}_p$ is a circle bundle over a closed $4n$-dimensional integral symplectic manifold. We use Wodzicki-Chern-Simons forms on the loop space $L\overline{M}_p$ to prove that $π_1({\rm Isom}(\overline{M}_p))$ is infinite for $|p| \gg 0.$ We also give the first high dimensional examples of nonvanishing Wodzicki-Pontryagin forms.

Theorems & Definitions (33)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• Proposition 3.5
• Proposition 4.1
• proof