Geodesic
Multiplications on a Space with Finitely Many Non-Vanishing Homotopy Groups
Canadian journal of mathematics, Tome 24 (1972) no. 6, pp. 1052-1062

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Copeland [1] proved that if X is a CW complex then the set of homotopy classes of multiplications is in 1-1 correspondence with the loop [X ∧ X, X], But in general [X ∧ X, X] is very hard to compute.Here we study the problem of finding how many different (up to homotopy) multiplications can be put on a space with finitely many non-vanishing homotopy groups. We reduce this problem to computing quotients of certain cohomology groups and to determining the primitive elements (Theorem 3.6). Our approach to this problem uses Postnikov decompositions and multiplier arguments. 
Cheng, Chao-Kun. Multiplications on a Space with Finitely Many Non-Vanishing Homotopy Groups. Canadian journal of mathematics, Tome 24 (1972) no. 6, pp. 1052-1062. doi: 10.4153/CJM-1972-107-4
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