We prove that a space M with Disjoint Disk Property is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. This implies that the product M × I² of a space M with the disk is a Q-manifold if and only if M × X is a Q-manifold for some C-space X. The proof of these theorems exploits the homological characterization of Q-manifolds due to Daverman and Walsh, combined with the existence of G-stable points in C-spaces. To establish the existence of such points we prove (and afterward apply) homological versions of the Brouwer Fixed Point Theorem and of Uspenskij's Selection Theorem.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc77-0-1,
author = {Taras Banakh and Robert Cauty},
title = {A homological selection theorem implying a division theorem for Q-manifolds},
journal = {Banach Center Publications},
volume = {75},
year = {2007},
pages = {11-22},
zbl = {1144.57020},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc77-0-1}
}
Taras Banakh; Robert Cauty. A homological selection theorem implying a division theorem for Q-manifolds. Banach Center Publications, Tome 75 (2007) pp. 11-22. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc77-0-1/