A homological selection theorem implying a division theorem for $Q$-manifolds
Volume 77 / 2007
Banach Center Publications 77 (2007), 11-22
MSC: 54C65, 55N10, 55M20, 54H25, 57N20, 57N75.
DOI: 10.4064/bc77-0-1
Abstract
We prove that a space $M$ with Disjoint Disk Property is a $Q$-manifold if and only if $M\times X$ is a $Q$-manifold for some $C$-space $X$. This implies that the product $M\times I^2$ of a space $M$ with the disk is a $Q$-manifold if and only if $M\times X$ 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.
