Multiplicative properties of power maps. II

McGibbon, C. A.

doi:10.1090/S0002-9947-1982-0675065-6

Multiplicative properties of power maps. II
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by C. A. McGibbon

Trans. Amer. Math. Soc. 274 (1982), 479-508

DOI: https://doi.org/10.1090/S0002-9947-1982-0675065-6

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Abstract:

The notions of $A_n$-maps and $C_n$-forms can be regarded as crude approximations to the concepts of homomorphisms and commutativity, respectively. These approximations are used to study power maps on connected Lie groups and their localizations. For such groups the power map $x \mapsto {x^n}$ is known to be an $A_2$-map if and only if $n$ is a solution to a certain quadratic congruence. In this paper, $A_3$-power maps are studied. For the Lie group Sp(l) it is shown that the $A_3$-powers coincide with solutions which are common to the quadratic congruence, mentioned earlier, and another cubic congruence. Other Lie groups, when localized so as to become homotopy commutative, are also shown to have infinitely many $A_3$-powers. The proofs reflect the combinatorial nature of the obstructions involved.

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