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

Wagner, Neal R.

doi:10.1090/S0002-9947-1974-0358781-0

A continuity property with applications to the topology of $2$-manifolds
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Trans. Amer. Math. Soc. 200 (1974), 369-393 Request permission

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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