Hopf construction map in higher dimensions (Q2493417)

In this paper, the author studies the zero set of the Hopf construction map \(F_n : A_n\times A_n \to A_n \times A_0\) given by \(F_n (x, y) = (2xy, | y|^2 - |x|^2)\) for \(n >3\) where \(A_n\) is the Cayley-Dickson algebra of dimension \(2^n\) over the real numbers. Define \(X_n^\infty = \{(x, y) \in A_n\times A_n\mid F_n(x, y) = (0, 0)\}\) and for \(r\) a nonnegative real number \((x, y) \in X^r_n\) if and only if \(xy = 0\) and \(\| x\| = \| y\| =r\). It is clear that for \(r>0\) and \(s>0\) real numbers, \(X^r_n\) is homeomorphic to \(X^s_n\). Define \(X_n := X^1_n\). The set \(X_n\) shows up in some important problems in algebraic topology: (1) Cohen's approach to the Arf invariant one problem. (2) Adem-Lam construction of normed and non-singular bilinear maps. In this paper, he shows that for \(n\geq 4\), \(X_n\) is related to some Stiefel manifolds; using the algebra structure in \(A_{n+1}\) he constructs the chain of inclusions \[ X_n\subset W_{2^{n-1}-1,2}\subset V_{2^n-2,2} \subset V_{2^n-1,2} \] where \(V_{m,2}\) and \(W_{m,2}\) denote the real and complex Stiefel manifolds of 2-frames in \(\mathbb R^m\) and \(\mathbb C^m\), respectively. In \S3 he shows that one can attach to every element in \(W_{2^{n-1},2}\) in a canonical way, an eight-dimensional vector subspace of \(A_{n+1}\) and that, only for the elements in \(X_n\), such vector subspace is isomorphic, as algebra, to \(A_3 = \emptyset\) (the octonions). In \S4 he describes \(X_n\) as a certain type of algebra monomorphisms from \(A_3 = \emptyset\) to \(A_{n+1}\) for \(n\geq 4\). This paper is a sequel to the author's previous one [\textit{G. Moreno}, Bol. Soc. Mat. Mex. 4, No. 3, 13--28 (1998; Zbl 1006.17005)] and use is made of the results of [\textit{R. D. Schafer}, Am. J. Math. 76, 435--446 (1954; Zbl 0059.02901)].