Vol. 37, No. 3, 1971

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A study of H-spaces via left translations

Robert Anthony Nowlan

Vol. 37 (1971), No. 3, 779–794

H-spaces are examined by studying left translations, actions and a homotopy version of left translations to be called homolations. If (F,m) is an H-space, the map s : F FF given by s(x) = Lx, i.e. s(x) is left translation by x, is a homomorphism if and only if m is associative. In general, s is an An-map if and only if (F,m) is an An+1 space. The action r : FF × F F is given by r(φ,x) = φ(x). The map s respects the action only of left translations. In general, s respects the action of homolations up to higherorder homotopies. Each homolation generates a family of maps to be called a homolation family. Denoting the set of all homolation families by H(F),s : F FF factors through F H(F) and this latter map is a homotopy equivalence.

Mathematical Subject Classification
Primary: 55D45
Received: 6 April 1970
Revised: 9 July 1970
Published: 1 June 1971
Robert Anthony Nowlan