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Multiplications on a Space with Finitely Many Non-Vanishing Homotopy Groups

Published online by Cambridge University Press:  20 November 2018

Chao-Kun Cheng
Chao-Kun Cheng*
Affiliation:
State University College at Potsdam, Potsdam, New York
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 6 , December 1972 , pp. 1052 - 1062
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Copeland, A. H., Jr., Binary operations on sets of mapping classes, Michigan Math. J. 6 (1969), 7–23.Google Scholar
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6. Zabrodsky, A., Homotopy associativity and finite CW complexes, Topology 9 (1970), 121–128.Google Scholar