Absolute extensors in dimension n and dimension-raising n-soft maps. (English. Russian original) Zbl 0572.54012

Dranishnikov, A. N.

Russ. Math. Surv. 39, No. 5, 63-111 (1984); translation from Usp. Mat. Nauk 39, No. 5(239), 55-95 (1984).

This paper introduces the reader to the theory of absolute extensors in dimension n, AE(n), and n-soft maps. A map \(f: X\to Y\) is said to be soft with respect to the pair (Z,A) if for any map \(g: Z\to Y\) and any continuous lifting of the map over A, \(h: A\to X\), there is a continuous lift \(H: Z\to X\). A map \(f: X\to Y\) is said to be n-soft if it is soft with respect to every pair of the type ”paracompactum of dimension n and closed subspace”. Theorems: For any natural number n there is an n- dimensional compactum and an (n-1)-soft map of the compactum onto the Hilbert cube. An (n-1)-soft map of an n-dimensional compactum onto the Hilbert cube cannot be made n-soft. In fact, any non-constant n-soft map of a connected n-dimensional compactum is cell-like. An n-soft map of an n-dimensional compactum onto the n-dimensional cube is a homeomorphism. A compactum X belongs to AE(n) if and only if X is an n-invertible image of the universal Menger space \(M_ n\).

Reviewer: D.C.Wilson

MSC:

54C55	Absolute neighborhood extensor, absolute extensor, absolute neighborhood retract (ANR), absolute retract spaces (general properties)
54F45	Dimension theory in general topology
54C10	Special maps on topological spaces (open, closed, perfect, etc.)

Keywords:

dimension raising mappings; absolute extensors in dimension n; n-soft maps; Hilbert cube; connected n-dimensional compactum; n-invertible image of the universal Menger space