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

Voir la notice de l'article provenant de la source Cambridge University Press

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
@article{10_4153_CMB_1968_079_0,
     author = {Hoo, C. S.},
     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/}
}
TY  - JOUR
AU  - Hoo, C. S.
TI  - A Generalization of a Theorem of Hilton
JO  - Canadian mathematical bulletin
PY  - 1968
SP  - 663
EP  - 669
VL  - 11
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-079-0/
DO  - 10.4153/CMB-1968-079-0
ID  - 10_4153_CMB_1968_079_0
ER  - 
%0 Journal Article
%A Hoo, C. S.
%T A Generalization of a Theorem of Hilton
%J Canadian mathematical bulletin
%D 1968
%P 663-669
%V 11
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1968-079-0/
%R 10.4153/CMB-1968-079-0
%F 10_4153_CMB_1968_079_0

[1] 1. Arkowitz, M., The generalized Whitehead product,. Pac. J. Math. 12 (1962) 7-23. Google Scholar

[2] 2. Arkowitz, M., Homotopy products for H-spaces, Mich. Math. J. 10 (1963) 1-9. Google Scholar

[3] 3. Arkowitz, M., Commutators and cup products, 111. J. Math. 8 (1964) 571-581. Google Scholar

[4] 4. Berstein, I. and Ganea, T., Homotopical nilpotency, 111. J. Math. 5 (1961) 99-130. Google Scholar

[5] 5. Berstein, I. and Hilton, P. J., On suspensions and comultiplications, Topology 2 (1963) 73-82. Google Scholar

[6] 6. Hilton, P.J., Note on a theorem of Stasheff, Bull. Pol. Acad. Sci. 10 (1962) 127-131. Google Scholar

[7] 7. Hilton, P.J., Homotopy theory and duality (Gordon and Breach, New York, 1965). Google Scholar

[8] 8. Hoo, C. S., A note on a theorem of Ganea, Hilton and Peterson. Proc. Amer. Math. Soc. 19 (1968) 909-911. Google Scholar

[9] 9. Stasheff, J., On homotopy Abelian H-spaces, Proc. Camb. Phil. Soc. 57 (1961) 734-745. Google Scholar

Cité par Sources :