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Abstract: Let M and N be even-dimensional oriented real manifolds, and u a smooth mapping from M to N. A pair of complex structures on M and N is called u-compatible if the mapping u is holomorphic with respect to these structures. The quotient of the space of u-compatible pairs of complex structures by the group of u-equivariant pairs of diffeomorphisms of M and N is called the moduli space of u-equivariant complex structures.
The paper contains a description of the fundamental group G of this moduli space in the following case: N = CP1, M is a hyperelliptic genus g curve in CP2 given by the equation y2 = Q(x), where Q is a generic polynomial of degree 2g+1, and u(x,y) = y2. The group G is the kernel of several (equivalent) actions of the braid-cyclic group BC2gon 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. G2g is a subgroup of index (2g+1)2g-2 in BC2g.
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