On divisibility in shape theory.

Abstract

Given two shape morphisms F,G:X→Y , where X and Y are compacta, one declares F to be a divisor of G provided for any compactum Z and any shape morphism U:X→Z if F factors as F=F 1 ∘U , then G factors as G=G 1 ∘U . On the other hand, if Sh(X,Y) is a group, then F being a divisor of G ought to mean that G=mF for some integer m . In particular, if Y=S n is the n -sphere, then Sh(X,S n )=[X,S n ] can be given the structure of a group (the n th cohomotopy group) if the shape dimension of X is at most 2n−1 . Here is the main result of the paper. Theorem. If F,G:X→S n and the shape dimension of X is at most n , then F is the divisor of G iff G=mF for some integer m in the n th cohomotopy group of X.