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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
##### MSC:
 55Q50 $$J$$-morphism 55P35 Loop spaces
##### Keywords:
generalized J-homomorphism; sphere; loop space; Stiefel manifold