Real cobordism and greek letter elements in the geometric chromatic spectral sequence
Let \(BP{\mathbb R}\) denote the Real Brown-Peterson spectrum which is obtained from the real cobordism spectrum \(M{\mathbb R}\), localized at the prime \(2\), in a way similar to the usual Brown-Peterson spectrum \(BP\). In a previous paper by the authors [Topology 40, 317--399 (2001; Zbl 0967.55010)] both spectra \(M{\mathbb R}\) and \(BP{\mathbb R}\) were investigated extensively. In particular, a spectral sequence for \(BP{\mathbb R}\) of the Adams-Novikov type was constructed. In this paper the authors give an application of this spectral sequence concerning a comparison between \(BP{\mathbb R}\) with other known spectral sequences on the edge level. First recall the chromatic spectral sequence \[ E_1=\oplus_n \text{Ext}_{BP_*} (BP_*, BP_*Y_n) \Rightarrow \text{Ext}_{BP_*BP_*}(BP_*, BP_*) \] where \(Y_n\) denotes Ravenel's \(BP\)-local spectrum. Here note that its terminal term is precisely the \(E_2\)-term of the Adams-Novikov spectral sequence with \((\pi_*S^0)\sphat_2\) as a limit. Moreover one has two types of spectral sequences, called the geometric Adams-Novikov and geometric chromatic spectral sequences, connecting the above \(E_1\)-term with \((\pi_*S^0)\sphat_2\) such that \[ \oplus_n \text{Ext}_{BP_*}(BP_*, BP_*Y_n) \Rightarrow \oplus_n \pi_*Y_n \Rightarrow (\pi_*S^0)\sphat_2. \] Let \(\omega\) be an element of \(\pi_*Y_n\) such that it has non-trivial Hurewicz homomorphism image in \(BP_*Y_n\). Then the main result (Theorem 1.8 and 1.9) of the paper concerns necessary conditions for \(\omega\) to be a permanent cycle or a target of a differential in the geometric chromatic spectral sequence. They are too complicated to be described here. But from them, for example, one finds that if \(\alpha_k\) is a permanent cycle then \(k \not\equiv 3 \mod 4\) and if \(\gamma_k\) is a permanent cycle then \(k \not\equiv 5 \bmod 16\). The \(BP{\mathbb R}\) may be regared as a \({\mathbb Z}/2\)-equivariant spectrum. This paper presents a good example in which the usefulness of such a \({\mathbb Z}/2\)-equivariant spectrum in the calculation of the usual spectrum \(BP\) is indicated.
scientific article; zbMATH DE number 2153107