Čech homology characterizations of infinite dimensional manifolds. (English) Zbl 0538.57006

Daverman, Robert J.; Walsh, John J.

Am. J. Math. 103, 411-435 (1981).

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

Keywords:

infinite codimension; locally compact ANR; Q-manifold; disjoint disks property; disjoint Čech carriers property; cellular decomposition; decomposition space