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We prove that a spaceM 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 × I 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. It was noticed in [BR] that the famous Shchepin’s Characterization Theorem for Tikhonov cube [Sc] implies a surprising Theorem 1 (Division Theorem for Tikhonov cubes). If the product X×Y of two spaces is homeomorphic to a Tikhonov cube [0, 1] of uncountable weight τ , then X or Y is homeomorphic to [0, 1] . This theorem has essentially non-metric nature and is not true for the Hilbert cube Q = [0, 1]. A suitable counterexample can be found in [Sin]. Example 1 (Singh). There is a compact absolute retract S such that S× [0, 1] is homeomorphic to Q but S contains no closed ANR-subspace of dimension ≥ 2. Singh’s space S contains no topological copy of the 2-disk I and hence does not possess the Disjoint Disks Property. We recall that a space X has the Disjoint Disks 2000 Mathematics Subject Classification: 54C65, 55N10, 55M20, 54H25, 57N20, 57N75 The paper is in final form and no version of it will be published elsewhere.

A homological selection theorem implying a division theorem for Q-manifolds