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Relative moduli spaces of complex structures: an example  Yurii M. Burman
;  Date: 
4 Mar 1999  Subject:  Differential Geometry; Combinatorics  math.DG math.CO  Affiliation:  Independent University of Moscow  Abstract:  Let M and N be evendimensional oriented real manifolds, and $u:M o N$ be a smooth mapping. A pair of complex structures at M and N is called ucompatible if the mapping u is holomorphic with respect to these structures. The quotient of the space of ucompatible pairs of complex structures by the group of uequivariant pairs of diffeomorphisms of M and N is called a moduli space of uequivariant complex structures. The paper contains a description of the fundamental group G of this moduli space in the following case: $N = CP^1, M subset CP^2$ is a hyperelliptic genus g curve given by the equation $y^2 = Q(x)$ where Q is a generic polynomial of degree 2g+1, and $u(x,y) = y^2$. The group G is a kernel of several (equivalent) actions of the braidcyclic group $BC_{2g}$ on 2g strands. These are: an action on the set of trees with 2g numbered edges, an action on the set of all splittings of a (4g+2)gon into numbered nonintersecting quadrangles, and an action on a certain set of subgroups of the free group with 2g generators. $G_{2g} subset BC_{2g}$ is a subgroup of the index $(2g+1)^{2g2}$. Key words: Teichmüller spaces, LyashkoLooijenga map, braid group.  Source:  arXiv, math.DG/9903029  Services:  Forum  Review  PDF  Favorites 


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