On certain homotopy-homomorphic elements of \(\pi _{n+1}(X)\) (Q913307)

Let X be a topological space with a base point \(x_ 0\) and let \(\Omega(X)\) be the loop space of X at \(x_ 0\). We give \(\Omega(X)\) the constant loop at \(x_ 0\) as a base point. As is well-known there exists an isomorphism: \(\pi_{n+1}(X)\to \pi_ n(\Omega X)\). We identify elements of these groups by this isomorphism. Now let a, b be given integers and \(\mu: S^ n\times S^ n\to S^ n\) be a map of type (a,b), i.e. such that \(\mu(x,*)\) and \(\mu(*,y)\) are maps \(S^ n\to S^ n\) of degree a and b respectively. We call an element \(\alpha\) of \(\pi_{n+1}(X)\) a \(\mu\)-homomorphic element (or to be \(\mu\)-homomorphic) if and only if \(\alpha(\mu(x,y))= \omega(\alpha (m_ a(x))\), \(\alpha (m_ b(y)))\), where \(\omega\) denotes the usual multiplication in \(\Omega\) (X) and \(m_ a\) is a map: \(S^ n\to S^ n\) of degree a (in fact \(m_ a(x)= \mu(x,*)).\) In this note our purpose is to find an obstruction for determining whether an element is \(\mu\)-homomorphic. As a result we prove Theorem 1. For an element \(\alpha\) of \(\pi_{n+1}(X)\), \(\alpha\) is \(\mu\)- homomorphic if and only if \(\alpha_*(c(\mu))=0\) where \(c(\mu)\) denotes the Hopf construction as defined by \textit{I. M. James} [Ann. Math., II. Ser. 63, 191-247 (1956; Zbl 0071.170)]. An analogous problem has been considered in case of \(\pi_ 3(G)\) for compact connected Lie groups G and \((a,b)=(1,1)\) by \textit{H. Takahashi} [``Homomorphisms from \(S^ 3\) to compact Lie groups up to homotopy'', Bull. Nagaoka Univ. Tech. (to appear)]. Our obstruction defines a correspondence \(\chi: \pi_ n(\Omega (X))\to \pi_{2n}(\Omega(X))\). This correspondence \(\chi\) is not necessarily homomorphic. We prove Theorem 2. \(\chi\) is homomorphic if \(\Omega(X)\) is a homotopy commutative Hopf space under the usual multiplication.