Properties of comultiplications on a wedge of spheres
A comultiplication \(\varphi\) on a space \(X\) is a map \(\varphi : X \to X \vee X\) satisfying \(j \circ \varphi \simeq \Delta\) where \(j : X \vee X \to X \times X\) is the inclusion and \(\Delta : X \to X \times X \) is the diagonal. The authors study the set of comultiplications when \(X\) is a wedge of \(t\) spheres for \(t > 2\) building on their previous work for \(t = 2\) [\textit{M. Arkowitz} and \textit{D.-W. Lee}, ``Comultiplications on a wedge of two spheres'', to appear]. A comultiplication \(\varphi\) in this case is characterized by a perturbation \(P = (P_1, \dots, P_t)\) where \(P_j\) satisfies \(\varphi \circ k_j \simeq \iota_1 \circ k_j + \iota_2 \circ k_j + P_j\). Here \(k_j\) is the inclusion of the \(j\)th sphere in the wedge \(X\) and the \(\iota_1, \iota_2 : X \to X \vee X\) are the inclusions. By Hilton's Theorem, the \(P_j\) may be expressed in terms of Whitehead products. The authors consider the cases where each \(P_j\) involves only length \(\leq 2\) or length \(\leq 3\) Whitehead products and prove extensive results on homotopy associativity, commutativity, inversivity etc. in these cases. The techniques used are classical with Hopf-Hilton invariants playing a central role.
scientific article; zbMATH DE number 5710098