\(H\)-spaces as direct product factors of loop spaces.
Let \(X\) denote an H-space. Then there is a mapping, constructed by \textit{I. M. James} [Ann. Math. (2) 62, 170--197 (1955; Zbl 0064.41505)], \(r: \Omega \Sigma X \to X\) which is a homotopy inverse to the natural mapping \(\rho: X \to \Omega\Sigma X\), the adjoint of the identity. This allows us to present \(\Omega\Sigma X \simeq X\times Y\) where \(Y\) is the homotopy fibre of \(r\), and thus embedding \(X\) in a loop space. This is not usually an H-map, however. When \(X\) is an \(A_n\)-space, in the sense of \textit{J. D. Stasheff} [Trans. Am. Math. Soc. 108, 275--292, 293--312 (1963; Zbl 0114.39402)], there is a higher-dimensional analogue of James's mapping splitting the adjoint of the inclusion of the suspension into \(P_nX\), the \(n\)th higher projective space associated to \(X\), \(\rho_n: X \to \Omega P_n X\). The splitting implies that \(\Omega P_nX\) has the homotopy type of \(X\times Y_n\). Analyzing the projective space, one finds that the inclusion of \(X\) into \(\Omega P_nX\) is an \(A_k\)-map for \(k\leq n-1\), and that a particular space can be used for \(Y_n\), namely \(\Omega \Sigma^nX^{\wedge n+1}\). The goal of this note is to analyze the homotopy equivalence \[ f_n: X\times \Omega\Sigma^nX^{\wedge n+1} \to \Omega P_nX. \] In particular, since the spaces involved are loop spaces of suspensions, their cohomology over a field is understood and so explicit properties can be determined. The motivating question posed by the author is the image associated to a class \(x\in \widetilde{H}^*(X;k)\), an \(A_n\)-class, for which there is \(\widehat{x}\), a class in \(H^*(P_nX;k)\) with \(i^*(\widehat{x}) = s(x)\) (\(i: \Sigma X\to P_nX\)). Then \[ f_n(\sigma^*(\hat{x})) = x\otimes 1 + 1\otimes \sigma^*(q_n^*(\widehat{x})), \] where \(q_n\) is a higher Hopf construction.
scientific article; zbMATH DE number 1961615
scientific article