Bi-\(Z_ 3\) equivariant maps between spheres (Q1179957)

A map \(f: S^{2k+1}\times S^{2n+1}\to S^{2n+1}\) is said to be bi- \(Z_ p\) equivariant if \(f(a,\cdot)\) and \(f(\cdot,b)\) are \(Z_ p\) equivariant, respectively, for arbitrary \(a\in S^{2k+1}\) and \(b\in S^{2n+1}\). Let \(L^ n(p)\) be a standard lens space. By the method of the real and complex Adams operations, the author proves that if \(k\not\equiv 0 \mod 4\), then \(\tilde J(L^ k(3))\cong \widetilde {KO}(L^ k(3))\). If \(x\neq 0\), and \(x=p^ sq^ t\cdots\) is the factorization of \(x\) into prime powers, we put \(v_ p(x)=s\). Then the author shows that, for \(k\not\equiv 0 \mod 4\), if there exists a bi-\(Z_ 3\) equivariant map: \(S^{2k+1}\times S^{2n+1}\to S^{2n+1}\), then \(v_ 3(n+1)\geqq [k/2]\).