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

V.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 . ecent 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• act 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. ⁎ 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 ) .