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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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