Some steps on short bridges: Non-metrizable surfaces and CW-complexes
Algebraic Topology
2012-10-02 v2
Abstract
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.
Cite
@article{arxiv.1103.3710,
title = {Some steps on short bridges: Non-metrizable surfaces and CW-complexes},
author = {Mathieu Baillif},
journal= {arXiv preprint arXiv:1103.3710},
year = {2012}
}
Comments
12 pages, 2 figures. V2: New figure added, some typos corrected, and Referee's remark included