$I$–adic towers in topology

Abstract

A large variety of cohomology theories is derived from complex cobordism $M{U}^{\ast }\left(-\right)$ by localizing with respect to certain elements or by killing regular sequences in $M{U}_{\ast }$. We study the relationship between certain pairs of such theories which differ by a regular sequence, by constructing topological analogues of algebraic $I$–adic towers. These give rise to Higher Bockstein spectral sequences, which turn out to be Adams spectral sequences in an appropriate sense. Particular attention is paid to the case of completed Johnson–Wilson theory $Ê\left(n\right)$ and Morava $K$–theory $K\left(n\right)$ for a given prime $p$.