Dyer-Lashof operations as extensions, and an application to H_*(BU)¿

Algebraic topology is concerned with the algebraic structure associated to topological spaces. There are algebraic operations $Q^k$, called Dyer-Lashof operations, that act on the homology of highly structured spaces. We explore a connection between these operations and $\Ext$ groups between unstable modules over the Steenrod algebra. This allows us to make calculations in the stable world, which is often easier. By using a purely algebraic spectral sequence developed by Kuhn and McCarty, along with these $\Ext$ groups, one can obtain information on how the $Q^k$ act on $H_*(\infloop X)$ for connective spectra $X$. The $\Ext$ groups are still not easy, but as an application of our method, we show how to calculate the $Q^k$ when $X=\Sigma^2 ku$ which has $H_*(\infloop X) = H_*(BU)$, obtaining an action of the Dyer-Lashof algebra that was previously shown by Kochman and Priddy.