Operations and cooperations in \(Im(J)\)-theory (Q1925117)

Let \(A\) denote the spectrum of connected \(\text{Im} (J)\)-theory at an odd prime. The main theorem is a splitting \(A\wedge A\simeq A\wedge\Sigma^{qn-1} X_K(n)\), taken over \(n\equiv 0\bmod p\). Here \(X_K(n)\) is the cofibre of a map \(K(n)\to K(n-1)\) of integral Brown-Gitler spectra. This of course yields a determination of \(A^*A\) and \(A_*A\) from \(A^* (X_K(n))\) and \(A_*(X_K(n))\). There is an isomorphism \(A^j(X_K(n)) \approx A_{j-3} (X_K(n))\). A prescription for calculating \(A^*(X_K(n))\) is given, but a comprehensive result is postponed until needed.