On real-oriented Johnson-Wilson cohomology
In 1985 Landweber defined an Atiyah-Real analogue \(M\mathbb{R}\) of the complex-cobordism spectrum \(MU\). It is an \(RO(\mathbb{Z}/2)\)-graded \(\mathbb{Z}/2\)-equivariant spectrum. S. Wilson asked the question if the Johnson-Wilson spectrum \(BP\langle n\rangle\) has a real analogue \(BP\mathbb{R}\langle n\rangle\) whose coefficient ring is the quotient of \(M\mathbb{R}_*\) by all elements ``not related to the generators \(v_0,\dots, v_n\)'' (recall that \(BP\langle n\rangle_*= \mathbb{Z}_{(2)}[v_1, v_2,\dots, v_n]\)). Using the techniques of Elmendorf-Kriz-Mandell-May, \(BP\langle n\rangle\) can be obtained from \(MU\) by killing elements of a suitable ideal \(I_n\) in \(MU_*\). There is an embedding \(MU_*\to M\mathbb{R}_*\), and the surprising fact is that the spectrum \(BP\mathbb{R}\langle n\rangle\) obtained from \(M\mathbb{R}\) by killing \(I_n\) does the job. The paper consists of the calculation of \(BP\mathbb{R}\langle n\rangle\). The author studies the Tate diagram of the \(\mathbb{Z}/2\)-spectrum \(E= BP\mathbb{R}\langle n\rangle\) \[ \begin{matrix} E\mathbb{Z}/2_+\wedge E &\longrightarrow & E &\longrightarrow &\widetilde{E\mathbb{Z}/2}\wedge E\\ @V\simeq VV \downarrow &&\downarrow\\ E\mathbb{Z}/2_+\wedge F(E\mathbb{Z}/2_+, E) &\longrightarrow &F(E\mathbb{Z}/2_+, E) &\longrightarrow &\widetilde{E\mathbb{Z}/2}\wedge F(E\mathbb{Z}/2_+,E)\end{matrix} \] induced from the cofiber sequence \[ E\mathbb{Z}/2_+\to S^0\to \widetilde{E\mathbb{Z}/2}. \] The coefficients of the Borel homology \(E\mathbb{Z}/2_+\wedge E\), the Borel cohomology \(F(E\mathbb{Z}/2_+,E)\) and the Tate spectrum \(\widetilde{E\mathbb{Z}/2}\wedge F(e\mathbb{Z}/2_+, E)\) are computed using spectral sequences, while \(\widetilde{E\mathbb{Z}/2}\wedge E\) is identified using geometric methods. The coefficients of \(E\) can be deduced from this data.
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