zbMATH — the first resource for mathematics

A proof of the theorem characterizing the generalized J-homomorphism. (English) Zbl 0695.55010
Homotopy theory and related topics, Proc. Int. Conf., Kinosaki/Japan 1988, Lect. Notes Math. 1418, 95-104 (1990).
[For the entire collection see Zbl 0685.00018.]
Let \(S^ n\) be the unit sphere, \(\Omega^ k S^ n\) for\(k\neq n\) the \(k\)-fold loop space of \(S^ n\) which is identified with a space of based maps from \(S^ k\) to \(S^ n\). Let \(V_{n,k}\) be the Stiefel manifold. One identifies \(V_{n,k}\) with a space of normed linear maps from \(\mathbb{R}^ k\) to \(\mathbb{R}^ n\). Then \(V_{k,n}\), acting on \(S^ k\) as \(\mathbb{R}^ k\) with a point at infinity, is considered as a subspace of \(\Omega^ k S^ n\). This defines an inclusion \(j_{n,k} : V_{n,k} \to \Omega^ k S^ n\). The induced map in homotopy is called the generalized \(J\)-map and is denoted by \[ J_{n,k} : \left[X,V_{n,k}\right] \to \left[X,\Omega^ k S^ n\right] \approx \left[\Sigma^ k X,S^ n\right]. \] One denotes by \(\partial : \left[\Sigma X,V_{n,k}\right] \to \left[X,S^{n-k-1}\right]\) for \(k+1\leq n\) the connecting map induced from the canonical fibration \(p : V_{n,k+1} \to V_{n,k}\). Then the main result of the paper is:
Theorem. There is a commutative diagram up to sign: \[ \begin{matrix} \left[\Sigma X, V_{n,k}\right] & @>{\quad J_{n,k} \quad}>> & \left[\Sigma^{k+1} X,S^ n\right] \\ \mathstrut_\partial \searrow && \nearrow \mathstrut_{\Sigma^{k+1}} \\ & \left[X, S^{n-k-1}\right] \end{matrix} \] In the case of \(X=S^ r\), various proofs are known [I.M. James, The topology of Stiefel manifolds (1976; Zbl 0337.55017)]. The purpose of the present note is to give a homotopy-theoretic proof using a method different from that of B. Gray [J. Lond. Math. Soc., II. Ser. 16, 124-130 (1977; Zbl 0396.55014)].
Reviewer: He Baihe
55Q50 \(J\)-morphism
55P35 Loop spaces