Let \(E(n)\) be the Johnson-Wilson spectrum at the prime 2. This is an algebra over the complex cobordism spectrum \(MU\) with coefficients \[ E(n)^* = \mathbb{Z}_{(2)} [v_1, \dots, v_{n-1}, v_n^{\pm 1} ] \] with \(v_k\) in cohomological degree \(-2(2^k -1)\). One can equip this spectrum \(E(n)\) with the structure of a genuine \(C_2\)-equivariant spectrum denoted \(\mathbb{ER}(n)\). This is analogous to how the complex conjugation action on \(MU\) can be used to construct the genuine \(C_2\)-equivariant spectrum \(\mathbb{MR}\). Forgetting the action on spectrum \(\mathbb{ER}(n)\) returns \(E(n)\). The \(C_2\)-fixed points of \(\mathbb{ER}(n)\) give a spectrum called \(ER(n)\). These spectra are higher chromatic versions of well-known spectra: \(ER(1)\) is real \(K\)-theory \(KO_{(2)}\) and \(\mathbb{ER}(1)\) is Atiyah’s Real \(K\)-theory \(\mathbb{KR}_{(2)}\).
The main result of this paper is a calculation of \(ER(n)^*( \mathbb{CP}^\infty )\) for all \(n\) using a Bockstein spectral sequence: \[ E_1^{i,j} = E(n)^{i \lambda(n)+j-i}(X) \implies ER(n)^{j-i}(X) \] where \(\lambda(n)=2(2^n-1)^2-1\). Furthermore the author obtains a ring description of \(ER(n)^*( \mathbb{CP}^\infty )\) in terms of generators and relations. This paper is part of a larger project to calculate the \(ER(n)\)-cohomology of basic spaces.