Subgroups of \(\pi_* (L_2T(1))\) at the prime two (Q1890211)

Let \(BP\) denote the Brown-Peterson spectrum for the prime \(2\), \(T(1)\) the Ravenel spectrum which is characterized by \(BP_*(T(1))= BP_* [t_1] \subseteq BP_*BP\), and let \(L_2\) denote the Bousfield localization functor with respect to \(G = v^{-1}_2BP\). This paper is concerned with the study of \(\pi_*(L_2(T(1)))\) by means of its Adams-Novikov spectral sequence. This spectral sequence has \(E_2\) term \(H^*(v^{-1}_2 BP_* [t_1])\), where \(H^*(\enskip)\) denotes \(Ext^*_{G*G}(G_*, \;)\), and this \(E_2\) term is studied using the chromatic spectral sequence which converges to it, whose \(E_2\) term is \(H^*(M^*_0 [t_1])\) for certain \(BP_*BP\) comodules \(M^*\), as described by [\textit{D. C. Ravenel}, Complex and Cobordism and Stable Homotopy Groups of Spheres. Pure and Applied Mathematics, Vol. 121. Orlando etc.: Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers) (1986; Zbl 0608.55001)]. The authors assemble information about this chromatic \(E_2\) term, including a computation of \(H^s (M^2_0 [t_1])\) for \(s > 4\), and then use this to study the Adams-Novikov spectral sequence of \(L_2 (T(1))\) by comparison with that of the related spectrum \(L_2 (T(1)) \bigwedge W\), where \(W\) is the cofibre of the localization map \(V \to L_1(V)\) and \(V\) is the cofibre of \(S^0 \to SQ\). The \(E_\infty\) term of the Adams-Novikov spectral sequence of this spectrum is identified and it is shown that \(E_4 = E_\infty\) in the case. The map \(\eta: W \to \Sigma V \to S^2\) induces a map of spectral sequences which is shown to be an isomorphism, hence showing that \(E_4 = E_\infty\) for \(L_2( T(1))\) also. From this the existence of certain explicitly described subgroups of \(\pi_* (L_2(T(1)))\) is deduced.