Maps between small Hopf spaces (Q2709728)

Let \(X\) be a finite \(H\)-space. It is well known that, under mild hypotheses, the set of homotopy classes of maps \([A, X]\) is an algebraic loop for any \(A\). If both \(A\) and \(X\) are \(H\)-spaces with no more than three cells, then it is known that \([A, X]\) is actually a group. There are \(15\) such \(H\)-spaces, which the authors list. Generally, these \(H\)-spaces admit more than one multiplication. Different multiplications on \(X\) generally lead to different group structures on \([A, X]\). For the \(225\) corresponding sets \([A, X]\), the authors determine the group structure, taking into account the choice of multiplication on \(X\) in their descriptions. Cases well known or previously considered elsewhere, such as \(A = S^n\), are omitted. The case in which \(A = X\) was considered in [\textit{M. Mimura} and \textit{H. Ōshima}, J. Math. Soc. Japan 51, No. 1, 71-92 (1999; Zbl 0931.55005)]. The (new) results are presented in a sequence of tables, which run across several pages. NEWLINENEWLINENEWLINETheir strategy is to use a ``Blakers-Massey'', or cofibration sequence, approach, by which \([A, X]\) is displayed as the middle term in a short exact sequence of groups. Then a rather arcane and close discussion identifies the first and last terms, together with the extension, for each of the possible cases. Parts of this discussion rely heavily on previously-known homotopy computations, such as Toda's computations of \(\pi_r(S^n)\) or those of [\textit{M. Mimura} and \textit{H. Toda}, J. Math. Kyoto Univ. 3, 217-250 (1964; Zbl 0129.15404)].