On the extension problem in the Adams spectral sequence converging to \(BP_* (\Omega^2 S^{2n+1})\) (Q2731059)

\textit{D. C. Ravenel} [Complex cobordism and stable homotopy groups of spheres, Pure Appl. Math. 121 (1986; Zbl 0608.55001); Nilpotence and periodicity in stable homotopy theory, Ann. Math. Stud. 128 (1992; Zbl 0774.55001)] computed the Adams spectral sequence converging to \(BP_*(\Omega^2 S^{2n+1})\) and also the \(E_\infty\)-term. Then he gave [Forum Math. 5, No. 1, 1-21 (1993; Zbl 0770.55009)] the following conjecture about the extension: NEWLINENEWLINENEWLINERavenel conjecture: \(BP_*(\Omega^2 S^{2n+1})= E(x_0)\otimes BP_*[y_i: i>0] /L\), where \(L\) is generated by the homogeneous components of the formal group law sum expression \(\sum^F_{0<j<i} v_jy_{i-j}^{p^j}\).NEWLINENEWLINENEWLINEThe author of this paper proves that there should be a nontrivial extension and he studies the \(BP_*BP\) comodule structures on the polynomial algebras which are related with \(BP_*(\Omega^2S^{2n+1})\).NEWLINENEWLINENEWLINEConcretely the following results are proved:NEWLINENEWLINENEWLINETheorem 6. If \(M_0=BP_*[y_i: i>0]\) then \(M_0\) is a left \(BP_*\)-module with a left \(BP_*\)-linear map \(\psi:M_0\to BP_*BP \otimes_{BP_*}M_0\), NEWLINE\[NEWLINE\sum_{j>0}^F y_j@>\psi>> \sum_{i\geq 0,j> 0}^Ft_i\otimes y_j^{p^i}NEWLINE\]NEWLINE which is co-unitary and associative, that is, \(M_0\) is a left \(BP_*BP\cong BP_*[t_1,t_2, \dots]\) comodule.NEWLINENEWLINENEWLINETheorem 7. If \(r_i=\sum_{0\leq j<i}v_j y_{i-j}^{p^j}\) then \(I=(r_1, r_2,\dots)\) is an invariant ideal, that is \(\psi(I)\subset BP_*BP \otimes_{BP_*} I\).NEWLINENEWLINENEWLINECorollary 8. \(M_n\), which is defined by \(BP_*[y_i:i> 0]/(r_1, \dots,r_n)\) is also a left \(BP_*BP\) comodule for each \(n\).NEWLINENEWLINENEWLINETheorem 10. There exist non-trivial extensions in the Adams spectral sequence covering to \(BP_*(\Omega^2 S^{2n+1})\). Furthermore, there exist various comodule structures on \(M_0\) or given comodule map, each comodule \(M_n\) is not uniquely determined from \(M_{n-1}\).