On dimensions modulo a compact metric ANR and modulo a simplicial complex

Abstract

V. V. Fedorchuk has recently introduced dimension functions Kdim ≤ K-Ind and L-dim ≤ L-Ind, where K is a simplicial complex and L is a compact metric ANR. For each complex K with a non-contractible join |K| * |K| (we write |K| for the geometric realisation of K), he has constructed first countable, separable compact spaces with K-dim < K-Ind.In a recent paper we have combined an old construction by P. Vopěnka with a new construction by V. A. Chatyrko, and have assigned a certain compact space Z(X, Y ) to any pair of non-empty compact spaces X, Y . In this paper we investigate the behaviour of the four dimensions under the operation Z(X, Y ). This enables us to construct examples of compact Fréchet spaces which have K-dim < K-Ind, L-dim < L-Ind, or K-Ind < |K|-Ind, and (connected) components of which are metrisable. In particular, given a natural number n ≥ 1, an ordinal α ≥ n, and any metric continuum C with L-dim C = n, we obtain • a compact Fréchet space X C,α such that L-dim X C,α = n, L-Ind X C,α = α, and each component of X C,α is homeomorphic to C.If L * L is non-contractible, or n = 1 and L is non-contractible, then C can be a cube [0, 1] m for a certain natural number m = m(n, L).2000 Mathematics Subject Classification. Primary 54F45.