Operations in the second quadrant Eilenberg-Moore spectral sequence

Abstract

The second quadrant Eilenberg-Moore spectral sequence has proved to be a very useful tool for computing the homology of fibre spaces. Though algebraic in nature, the spectral sequence has been given a purely geometric realization by Hodgkin, Rector [9], Smith [l I] and Heller. A consequence of their work is the existence of Steenrod operations in the spectral sequence. In this note we use these operations together with the naturality properties to introduce Dyer-Lashof type operations into the spectral sequence for fibre squares made up of infinite loop spaces. In particular, we define an action of the Dyer-Lashof operations on TorHeB(H*X, H* Y) where X, Y and B are infinite loop spaces and the module action is given by infinite loop space maps X-B and Y-B. We have, however, in the interest of compactness, omitted a cofmputational :esult from [l] which shows that these operations respect the tensor product splitting of Tor H.B(H*X, H*Y) given in those cases where the change of rings spectral sequence collapses.