Hereditarily aspherical compacta. (English) Zbl 0876.55002
A natural extension of the notion of asphericity is the concept of (strong) hereditary asphericity, introduced by R. J. Daverman [Topology Appl. 41, No. 3, 247-254 (1991; Zbl 0772.54026)]. Following his work and subsequent joint results with A. N. Dranishnikov [Ill. J. Math. 40, No. 1, 77-90 (1996; Zbl 0859.54021)] in which several important properties of these two classes of compacta were established, the following natural question was raised: Is the class of locally simply connected hereditarily aspherical compacta included in the class of strongly hereditarily aspherical compacta? In the present paper this question is answered in the affirmative, and in a strong sense – locally simply connected hereditarily aspherical compacta are shown to be ANR’s. The authors also introduce a property of shape asphericity and proceed to investigate it. As a corollary they get – among other interesting results – another proof of a theorem originally due to A. N. Dranishnikov and D. Repovš, to the effect that Kainian compacta (which they call compacta of perfect cohomological dimension one) are at most 2-dimensional [Topology Appl. 74, No. 1-3, 123-140 (1996; Zbl 0873.55001)].
Reviewer: D.Repovš (Ljubljana)
Keywords:
cell-like map; AE; compact metric space; asphericity; ANR; shape asphericity; cohomological dimensionReferences:
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