Abstract
Among the classical variants of the Prüfer surface, some are homotopy equivalent to a CW-complex (namely, a point or a wedge of a continuum of circles) and some are not. The obstruction comes from the existence of uncountably many ‘infinitesimal bridges’ linking two metrizable open subsurfaces inside the surface. We show that any non-metrizable surface that possesses such a system of infinitesimal bridges cannot be homotopy equivalent to a CW-complex. More than for the result on its own, we were motivated by trying to blend elementary techniques of algebraic and set-theoretic topology.
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Communicated by Jonathan Rosenberg.
For useful conversations, I would like to thank David Gauld, Sina Greenwood, and David Cimasoni.
I thank Alexandre Gabard for his numerous suggestions.
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Baillif, M. Some steps on short bridges: non-metrizable surfaces and CW-complexes. J. Homotopy Relat. Struct. 7, 153–164 (2012). https://doi.org/10.1007/s40062-012-0001-8
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DOI: https://doi.org/10.1007/s40062-012-0001-8