Homology operations and \(v _{n}\)-Bockstein operations (Q1873305)

This is a deep paper which sets out to determine a part of the Atiyah-Hirzebruch spectral sequence for Morava \(K\)-theory. In the \(\pmod p\) homology of \(\Omega^\ell\Sigma^\ell X\), there are three basic types of operations (Dyer-Lashof, dual Steenrod, and \(\lambda\)-operations). The author's main result yields a thorough analysis of the interactions of these operations. Using the stability of homology and the Snaith splitting in terms of \(D^{(\ell)}_j(X)\), see [\textit{V. P. A. Snaith}, J. Lond. Math. Soc. (2) 7, 577--583 (1974; Zbl 0275.55019)], the result is obtained up to certain undetermined coefficients in \(\mathbb F_p\). These are specifically worked out via calculations in \(B\Sigma_p\). Details are much too complex to be included here.