The Adams-Novikov \(E_2\)-term for \(\pi_*(L_2S^0)\) at the prime 2
The authors compute the \(E_2\)-term of the Adams-Novikov spectral sequence, based on the Johnson-Wilson theory \(E(2)\) for the prime \(p=2\), converging to the homotopy groups of \(L_2S^0\), the \(E(2)\)-localized sphere spectrum. To get to the \(E_2\)-term the authors proceed in the standard way, i.e.~they use the chromatic spectral sequence. As an input to this one needs to compute the Ext-groups \(H^*(M)= \text{Ext}^*_{E(2)_*E(2)}(M)\) for certain \(E(2)_*E(2)\)-comodules \(M=M^0_0, M^1_0\) and \(M^2_0\). The term \(H^*(M^0_0)\) is given by a single copy of the rationals \({\mathbb Q}\) in dimension \(0\), and \(H^*(M^1_0)\) has been computed by \textit{H. R. Miller}, \textit{D. C. Ravenel} and \textit{W. S. Wilson} [Ann. Math. (2) 106, 469-516 (1977; Zbl 0374.55022)]. So the authors essentially need to compute \(H^*(M^2_0)\) where \(M^2_0={\mathbb Z}_{(2)}[v_1,v_2^{\pm 1}]/(2^\infty,v_1^\infty)\). In order to compute \(H^*(M^2_0)\) the authors start from \(H^*(E(2)_*/(2,v_1))\) which has been calculated by \textit{D. C. Ravenel} [Math. Z. 152, 287-297 (1977; Zbl 0338.55018)] and use the \(v_1\)-Bockstein spectral sequence and then the \(2\)-Bockstein spectral sequence to finally determine \(H^*(M^2_0)\). This strategy has been proposed by Miller, Ravenel and Wilson [loc. cit.], and it also has been used successfully for computing the corresponding terms in the case where \(p\) is an odd prime [the authors, Topology 41, No. 6, 1183-1198 (2002; Zbl 1007.55007) for \(p=3\) and \textit{K. Shimomura} and \textit{A. Yabe}, Topology 34, No. 2, 261-289 (1995; Zbl 0832.55011) for \(p>3\))]. The actual computation for \(p=2\) here however is much harder and more involved (as expected). The results show that the homotopy group \(\pi_i(L_2S^0)\) contains two copies of \({\mathbb Q}/{\mathbb Z_{(2)}}\) in case \(i=-4\) and one copy of \({\mathbb Q}/{\mathbb Z_{(2)}}\) in case \(i=-5\); moreover they show that \(\pi_i(L_2S^0)\) is finite in case \(i\) is different from \(0,-4\) and \(-5\).
scientific article; zbMATH DE number 1837381