Geodesic
A Generalization of a Theorem of Hilton
Canadian mathematical bulletin, Tome 11 (1968) no. 5, pp. 663-669

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Let f: A×B → X be a map. Let J(f): ∑(A ∧ B)→ ∑X be the map obtained from f by means of the Hopf construction. Let P(f) denote the space obtained from ∑X by attaching a cone on ∑(A ∧ B) by means of J(f). Let l: ∑X→P(f ) be the inclusion and T(l): X→ΩP(f) the adjoint of l. Let h1,: A1, → A, h2: B1,→ B be maps. Let c:ΩP(f)× ΩP(f) → ΩP(f) be the basic commutator. Then we prove that there exists a map ∑A1 × ∑B1 →P(f) with axes l∑(fi1 h1), l∑(fi2 h2) if and only if , where i1: A→ A × B and i2 B → A × B are the inclusions. This generalizes a result of Hilton. Also, by letting f be an H-space multiplication and h and h the identity maps, we obtain a well known criterion of Stasheff for an H-space to be homotopy-commutative. Finally, appropriate duals of these results are given. 
Hoo, C. S. A Generalization of a Theorem of Hilton. Canadian mathematical bulletin, Tome 11 (1968) no. 5, pp. 663-669. doi: 10.4153/CMB-1968-079-0
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     title = {A {Generalization} of a {Theorem} of {Hilton}},
     journal = {Canadian mathematical bulletin},
     pages = {663--669},
     year = {1968},
     volume = {11},
     number = {5},
     doi = {10.4153/CMB-1968-079-0},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-079-0/}
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