Refinable maps onto locally n-connected compacta

Abstract

In [4], J. Ford and J.W. Rogers introduced the notion of refinable maps and they proved that each refinable map from a continuum to a locally connected continuum is monotone [4, Corollary 1.2]. In [5, Theorem 2.2], we proved that each refinable map from a compactum to an FANR induces a shape equivalence. In this paper we shall prove that if a map r: X-* Y between compacta is refinable and FeLC1 (n^O), then r~\y)^ACn for each jeF. Moreover if Y is an ANR, then r is a CE-map. It is assumed that all spaces are metrizable and maps are continuous. A connected compactum is a continuum. A map /: X―> Y between compacta is an e-mapping, s>0, if / is surjective and diamf~1(y) 0 there is an s-mapping /: X^Y such that d(r,f)―swp{d(r(x), f(x))\x<BX}<s. Such a map / is calledan e-refinement of r. Note that every refinable map is surjective,every near homeomorphism is refinable and if there is a refinable map from a compactum I to a compactum Y, then X is F-like. But simple examples show that any converse assertions of them are not true. A space X is locally n-connected (XgLC71) if for each x^X and an open neighborhood U of x in X, there is an open set V with xeFct/ such that each map h: Sk-+V is null-homotopic in U for OSk^n, where Sk denotes the ^-sphere. A compactum X in the Hilbert cube Q is approximatively n-connected (X<eACu) if for each open neighborhood U of X in Q there is an open neighborhood VdU of X in Q such that each map h: Sk~^V is null-homotopic in U for O^k^n (see [2]). A map f:X―*Y between compacta is a CE-map if / is surjective and f~\y)is an FAR (see [2]) for each y^Y.