Suppose that P is a finite 2-polyhedron. We prove that there exists a PL surjective map f:Q → P from a fake surface Q with preimages of f either points or arcs or 2-disks. This yields a reduction of the Whitehead asphericity conjecture (which asserts that every subpolyhedron of an aspherical 2-polyhedron is also aspherical) to the case of fake surfaces. Moreover, if the set of points of P having a neighbourhood homeomorphic to the 2-disk is a disjoint union of open 2-disks, and every point of P has an arbitrarily small 2-dimensional neighbourhood, then we may additionally conclude that Q is a special 2-polyhedron.
@article{bwmeta1.element.bwnjournal-article-cmv86i2p231bwm,
author = {Du\v san Repov\v s and Arkady Skopenkov},
title = {Cell-like resolutions of polyhedra by special ones},
journal = {Colloquium Mathematicae},
volume = {84/85},
year = {2000},
pages = {231-237},
zbl = {0966.57007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv86i2p231bwm}
}
Repovš, Dušan; Skopenkov, Arkady. Cell-like resolutions of polyhedra by special ones. Colloquium Mathematicae, Tome 84/85 (2000) pp. 231-237. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv86i2p231bwm/
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