Some steps on short bridges: Non-metrizable surfaces and CW-complexes

Details Some steps on short bridges: Non-metrizable surfaces and CW-complexes Some steps on short bridges: Non-metrizable surfaces and CW-complexes

2011-03-18

arxiv.org/abs/1103.3710v2

Journal of Homotopy and Related Structures, vol. 6(2), 2011, pp.239-251

Mathematics

Algebraic Topology

12 pages, 2 figures. V2: New figure added, some typos corrected, and Referee's remark included

Scientific paper

Among the classical variants of the Pr\"ufer 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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