The inverses of an \(H\)-space
Let \(X\) be an \(H\)-space with multiplication \(\mu : X\times X \to X\). Then \(\mu\) induces a binary operation on the set of homotopy classes of maps \([A, X]\) for any space \(A\). If \(\mu\) is homotopy associative, then \([A, X]\) is a group (at least when \(A\) is a CW complex). In general, however, \(\mu\) need not be homotopy associative. Some \(H\)-spaces do not admit any homotopy associative multiplication -- \(S^7\) is the simplest example. Many \(H\)-spaces admit more than one multiplication -- often infinitely many -- and may admit both homotopy associative and non-homotopy associative ones. If \(\mu\) is not homotopy associative, then \([A, X]\) in general has the structure of a loop (in the algebraic sense). In this case, various notions commonly associated with a binary operation are less well understood, and their properties typically are highly dependent on the particular choice of \(\mu\). This paper focusses on inverses. Each element \(\alpha\) in the loop \([A, X]\) has a unique left inverse \(\alpha_L\) and a unique right inverse \(\alpha_R\), which in general are distinct. If \(\lambda\) and \(\rho\) are, respectively, the left and right inverses of the identity \(\iota \in [X, X]\), then \(\alpha_L = \lambda\circ\alpha\) and \(\alpha_R = \rho\circ\alpha\). The main question considered is the one concerning the order of the (left) inverse \(\lambda\), i.e., the smallest \(n\) for which \(\lambda^n = \iota\). The authors observe that \(\lambda^2 = \iota\) is a necessary, but not a sufficient, condition for homotopy associativity. They give an example of a multiplication on \(S^3\times S^3\times S^3\) for which \(\lambda\) has order \(6\). Various other basic properties concerning inverses are established. Some broader results are obtained by using a rational homotopy approach, similar to that used in [\textit{M. Arkowitz} and \textit{G. Lupton}, Math. Proc. Camb. Philos. Soc. 110, No. 1, 121-136 (1991; Zbl 0732.55008)]. Using this approach, the authors give conditions under which \(X\) admits infinitely many multiplications that share the same left inverse \(\lambda\) and necessary and sufficient conditions for \(X\) to admit a multiplication for which \(\lambda\) has infinite order.
scientific article; zbMATH DE number 1801574