Probing mapping class groups using arcs

Abstract

The action of the mapping class group of a surface on the collection of homotopy classes of disjointly embedded curves or arcs in the surface is discussed here as a tool for understanding Riemann's moduli space and its topological and geometric invariants. Furthermore, appropriate completions, elaborations, or quotients of the set of all such homotopy classes of curves or arcs give for instance Thurston's boundary for Teichmüller space or a combinatorial description of moduli space in terms of fatgraphs. Related open problems and questions are discussed.

21 pages, 6 figures, expanded version

References in corpus (5)

• Real Multiplication and noncommutative geometry
• Combinatorial Miller-Morita-Mumford classes and Witten cycles
• Closed/open string diagrammatics
• Arc Operads and Arc Algebras
• The Structure and Singularities of Arc Complexes

Cited by in corpus (7)

• Closed/open string diagrammatics
• Aspects of line operators of class S theories
• The second rational homology group of the moduli space of curves with level structures
• Pseudo-Anosov mapping classes not arising from Penner's construction
• Cluster algebras and triangulated surfaces. Part I: Cluster complexes
• Stable curves and screens on fatgraphs
• Noncommutative aspects of open/closed strings via foliations