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A continuity property with applications to the topology of $ 2$-manifolds


Author: Neal R. Wagner
Journal: Trans. Amer. Math. Soc. 200 (1974), 369-393
MSC: Primary 57A05
MathSciNet review: 0358781
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Abstract: A continuity property is proved for variable simply connected domains with locally connected boundaries. This theorem provides a link between limits of conformal mappings and of retractions. Applications are given to the space of retractions of a compact $ 2$-manifold $ {M^2}$, where it is shown that the space of deformations retractions is contractible and the space of nullhomotopic retrac tions has the same homotopy type as $ {M^2}$. Other applications include a proof that the space of retracts of $ {M^2}$ (with a natural quotient topology) is an absolute neighborhood retract, and a type of global solution to the Dirichlet problem.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0358781-0
Keywords: Absolute neighborhood retract, Carathéodory convergence, compact-open topology, conformal mapping, continuity property, cross section, Dirichlet problem, Fréchet convergence, homotopy equivalence, locally trivial fibre space, prime end of the first kind, retract, retraction, Schoenflies theorem, two-manifold, variable domain
Article copyright: © Copyright 1974 American Mathematical Society