On the \(\Lambda\)-module structures of \(\tau\)-homotopy groups of \(X\bigwedge S_+^{1,0}\) (Q1059303)

A \(\tau\)-complex X is a G-complex for \(G={\mathbb{Z}}/2{\mathbb{Z}}\), generated by \(\tau\). The author studies the (unstable) \(\tau\)-homotopy groups \(\pi_{p,q}(X\bigwedge S_+^{1,0})\). Let \(p\geq 1\) and \(q\geq 2\). If \(\pi_ k(X\times X,X\bigvee X)=0\) for each k, \(q+2\leq k\leq p+q+1\), then there exists an isomorphism of abelian groups \(\pi_{p,q}(X\bigwedge S_+^{1,0})\cong \pi_{p+q}(X)\oplus \pi_{q+1}(X)\). Let \(\pi^ s_{0,0}={\mathbb{Z}}[\rho]/(1-\rho^ 2)\). The \(\rho\)-action on \(\pi_{p,q}(X\bigwedge S_+^{1,0})\) is given by \(\rho \cdot (\alpha,\beta)=(-\alpha,\beta)\) for \(p\geq 2\) and \(\rho \cdot (\alpha,\beta)=(-\alpha,\alpha +\beta)\) for \(p=1\). Moreover, the author shows that the stable \(\tau\)-homotopy group \(\pi^ s_{p,q}(X\bigwedge S_+^{1,0})\) is additively isomorphic to \(\pi^ s_{p+q}(X)\) and \(\rho\) acts as -1 on \(\pi^ s_{p,q}(X\bigwedge S_+^{1,0})\), where (p,q)\(\in {\mathbb{Z}}\times {\mathbb{Z}}\).