Circular groups, planar groups, and the Euler class

2004/12/13 by Danny Calegari · 67 citations
Mathematics · #Advanced Algebra and Geometry #Algebra over a field #Artificial intelligence #Class (philosophy) #Combinatorics #Computer graphics (images) #Computer science #Euler's formula #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Planar #Pure mathematics #Topology (electrical circuits)

paper · doi:10.2140/gtm.2004.7.431

published in Geometry and topology monographs, 431-491 (Mathematical Sciences Publishers)

openalex publication_date 2004/12/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

ABSTRACT. We study groups of C 1 orientation–preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly orderable groups. We show that every such group with a bounded orbit is circularly orderable, and show that certain generalized braid groups are circularly orderable. We also show that the Euler class of C ∞ diffeomorphisms of the plane is an unbounded class, and that any closed surface group of genus> 1 admits a C ∞ action with arbitrary Euler class. On the other hand, we show that Z⊕Zactions satisfy a homological rigidity property: every orientation–preserving C 1 action of Z⊕Zon the plane has trivial Euler class. CONTENTS

Citations

• Orderings of mapping class groups after Thurston
• Diffeomorphisms of the 2-sphere
• The Classification of Immersions of Spheres in Euclidean Spaces

Cited by

• The braided Ptolemy–Thompson group is finitely presented
• Small curvature laminations in hyperbolic 3–manifolds
• Cyclic order [wikipedia]