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

AbstractV.V. Fedorchuk has recently introduced dimension functions K-dim⩽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-dimC=n, we obtain•a compact Fréchet space XC,α such that L-dimXC,α=n, L-IndXC,α=α, and each component of XC,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)