Bing maps and finite-dimensional maps

Let X and Y be compacta and let f : X → Y be a k-dimensional map. In [5] Pasynkov stated that if Y is finite-dimensional then there exists a map g : X → double-struck I sign^k such that dim(f x g) = o. The problem that we deal with in this note is whether or not the restriction on the dimension of Y in the Pasynkov theorem can be omitted. This problem is still open. Without assuming that Y is finite-dimensional Sternfeld [6] proved that there exists a map g : X → double-struck I sign^k such that dim(f x g) = 1. We improve this result of Sternfeld showing that there exists a map g : X → double-struck I sign^k+1 such that dim(f x g) = 0. The last result is generalized to maps f with weakly infinite-dimensional fibers. Our proofs are based on so-called Bing maps. A compactum is said to be a Bing compactum if its compact connected subsets are all hereditarily indecomposable, and a map is said to be a Bing map if all its fibers are Bing compacta. Bing maps on finite-dimensional compacta were constructed by Brown [2]. We construct Bing maps for arbitrary compacta. Namely, we prove that for a compactum X the set of all Bing maps from X to double-struck I sign is a dense Gδ-subset of C(X,double-struck I sign).