Configuration models for moduli spaces of Riemann surfaces with boundary
type: Item
instance of: scholarly article
title: Configuration models for moduli spaces of Riemann surfaces with boundary
zbMATH Open document ID: 1125.32006
DOI: 10.1007/BF02960865
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 2007-08-29T00:00:00Z
review text: The author studies the moduli space \(\mathfrak{M}^{\bullet}_g(m,n)\) of Riemann surfaces of genus \(g\geq 0\) with \(n\geq 1\) incoming and \(m\geq 1\) outgoing boundary curves and with one marked point on each incoming curve. Such a surface is seen as a bordism between non-empty compact 1-manifolds. The author establishes a homeomorphism between \(\mathfrak{M}^{\bullet}_g(m,n)\) and a space \(\mathfrak{Rad}_h(m,n)\) consisting of configurations of \(h=2g-2+m+n\) pairs of radial slits distributed over \(n\) annuli, a slit being a radial cut from the outer boundary to some interior point. (Certain combinatorial conditions are required to insure genus \(g\) and \(m\) outgoing circles for the corresponding surface.) The association of a configuration to a surface of the type considered is called Hilbert uniformization, since the idea is already contained in David Hilbert's work. The author of the paper under review already used this method in earlier works. The space \(\mathfrak{Rad}_h(m,n)\) has a topology which makes it a noncompact manifold of dimension \(6g-6+3m+4n\), and the author gives a description of this space as the complement of a subcomplex in a finite cell complex. The author also studies the boundary structure obtained by degenerations of configurations. He describes a compactification of the moduli space where degenerate surfaces are added. This compactification is different from the Deligne-Mumford compactification of the usual moduli space of Riemann surfaces, since the degeneration here does not only consist of pinching curves, but also of degenerating handles to intervals. The author, in joint work with J. Abhau and E. Ehrenfried, used this compactification to compute the homology of the space \(\mathfrak{M}^{\bullet}_g(m,n)\). The author also mentions connections with operad structures conformal field theories.
reviewed by: Athanase Papadopoulos
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 30F15
Mathematics Subject Classification ID: 30F60
zbMATH DE Number: 5184538
zbMATH Keywords: Riemann surface
zbMATH Keywords: moduli space
zbMATH Keywords: slit
zbMATH Keywords: Hilbert uniformization
author: C.-F. Bödigheimer
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02960865
OpenAlex ID: W2052222835
description: scientific article; zbMATH DE number 5184538