Homotopy comultiplications on the \(k\)-fold wedge of spheres
The paper under review is the natural continuation of the works by \textit{M. Arkowitz} and \textit{D.-W. Lee} [Sci. China, Math. 54, No. 1, 9--22 (2011; Zbl 1229.55009); Topology Appl. 157, No. 9, 1607--1621 (2010; Zbl 1203.55007)]. The focus is on the cardinality of the set of homotopy comultiplications on the \(k\)-fold wedge of spheres \(Z=\bigvee_{i=1}^k\mathbb{S}^{n_i}\) for arbitrary and sufficiently large values of \(k\), where \(2\leq n_1<n_2<\dots<n_k\). The main ingredients are basic Whitehead products on \(Z\) and the Hopf-Hilton invariants. The authors develop formulas for three cases. The first formula gives the number of comultiplications, the second the number of comultiplications which are associative and the third the number of comultiplications which are commutative. The results are not simple even to state and involve many technical details and notation. We illustrate one of the cases, namely the number of comultiplications. As a consequence of one of the main results, Theorem 2.14, the authors show: Corollary 2.15. The number of comultiplications of \(Z=\mathbb{S}^{n_1}\vee \mathbb{S}^{n_2}\vee\dots \vee\mathbb{S}^{n_k}\) is \[ |\mathcal{C}(Z)|=\Pi_{w_2\in W_ 2^{*}}|\pi_{n_2}(\mathbb{S}^{h_{w_2}})|\times \Pi_{w_3\in W_3^{*}}|\pi_{n_3}(\mathbb{S}^{h_{w_3}})|\times \dots\times \Pi_{w_k\in W_k^{*}}|\pi_{n_k}(\mathbb{S}^{h_{w_k}})|. \] The sets \(W_i^{*}\) are certain basic Whitehead products which involve elements from the homotopy of each of the factors of \(Z\vee Z\) and the spheres \(\mathbb{S}^{h_{w_i}}\) are given by Hilton's formula for the description of the homotopy of \(Z\). Then two similar results concerning the cardinality of associative and commutative comultiplications are given. At the end the authors present some figures to analyse certain regularities and patterns of the commutative comultiplications.
scientific article; zbMATH DE number 7005612
scientific article