%0 Book Section
%T On divisibility in shape theory.
publisher Editorial Complutense
%D 1994
%U 84-7491-510-4
%@ https://hdl.handle.net/20.500.14352/60725
%X 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.
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