Abstract
By an AR(ANR), one understands a compact metric absolute retract (compact metric absolute neighborhood retract). It is well known that AR's contain arcs. One may ask whether every AR(ANR) of dimension n, n ≥2, contains a cell of dimension k, where k≥ 2. Likewise, one may ask whether every AR of dimension n, n ≥2, contains a proper AR(ANR) of dimension k, where k ≥2. Answers to questions of this type will enhance the understanding of the structure of AR's (ANR's). This chapter discusses the recent results and some open problems in this direction. It describes a general method of constructing decompositions of B3 such that the decomposition spaces are retracts of the suitable type. To avoid technicalities, these decompositions are described for E.
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