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On the disjoint \((0,n)\)-cells property for homogeneous ANR’s. (English) Zbl 0849.54026

Summary: A metric space \((X, \rho)\) satisfies the disjoint \((0, n)\)-cells property provided for each point \(x\in X\), any map \(f\) of the \(n\)-cell \(B^n\) into \(X\) and for each \(\varepsilon > 0\) there exist a point \(y\in X\) and a map \(g\in B^n\to X\) such that \(\rho (x, y)< \varepsilon\), \(\widehat {\rho} (f, g)< \varepsilon\) and \(y\not\in g(B^n)\). It is proved that each homogeneous locally compact ANR of dimension \(>2\) the disjoint \((0, 2)\)-cells property. If \(\dim X=n>0\), \(X\) has the disjoint \((0,n-1)\)-cells property and \(X\) is a locally compact \(LC^{n-1}\)-space then local homologies satisfy \(H_k (X, X-x) =0\) for \(k< n\) and \(H_n (X, X-x) \neq 0\).

MSC:

54F35 Higher-dimensional local connectedness
55M15 Absolute neighborhood retracts
54C55 Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
57P05 Local properties of generalized manifolds
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