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Čech homology characterizations of infinite dimensional manifolds. (English) Zbl 0538.57006
Let $$X$$ denote a locally compact ANR. It is shown that $$X$$ is a $$Q$$-manifold if and only if $$X$$ has the disjoint disks property and the disjoint Čech carriers property. This result can be regarded as the infinite- dimensional version of the known fact that when $$n\geq 5$$, a generalized n-manifold Y is a topological $$n$$-manifold if and only if $$Y$$ has the disjoint disk property; the proof relies on the result of H. Torunczyk [Fundam. Math. 106, 31-40 (1980; Zbl 0346.57004)] that $$X$$ is a $$Q$$-manifold if and only if $$X$$ has the disjoint $$k$$-cells property for each $$k\geq 0$$. Several applications are given to the problems of recognizing $$Q$$-manifolds among products, proper cell-like images of $$Q$$-manifolds, and sums of $$Q$$-manifolds. An example is also given of a cellular decomposition of $$Q$$ whose nondegenerate elements form a null-sequence, but whose decomposition space is not a $$Q$$-manifold.
Reviewer: R.Sher

##### MSC:
 57N20 Topology of infinite-dimensional manifolds 57N60 Cellularity in topological manifolds 58B05 Homotopy and topological questions for infinite-dimensional manifolds 57P99 Generalized manifolds 55N07 Steenrod-Sitnikov homologies
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