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The structure and singularities of quotient arc complexes
Author(s) -
Penner R. C.
Publication year - 2008
Publication title -
journal of topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.447
H-Index - 31
eISSN - 1753-8424
pISSN - 1753-8416
DOI - 10.1112/jtopol/jtn006
Subject(s) - mathematics , moduli space , riemann surface , quotient , gravitational singularity , pure mathematics , riemann sphere , boundary (topology) , modulo , mathematical analysis , combinatorics , geometry
A well‐known combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non‐empty boundary, there is an analogous complex QA( F ) consisting of equivalence classes of arcs in F connecting a given finite set of points in its boundary modulo diffeomorphisms of F pointwise fixing the boundary and any punctures. The main result of this paper is the determination of those complexes QA( F ) that are also spheres. This classification has consequences for Riemann's moduli space of curves via its known identification with a related quotient arc complex in the punctured case with no boundary. Namely, the essential singularities of the natural cellular compactification of Riemann's moduli space can be described.