Steenrod and Dyer-Lashof operations on 𝐵𝑈

Abstract

This paper describes a simple, fast algorithm for the computation of Steenrod and Dyer-Lashof operations on B U BU . The calculations are carried out in H ∗ ( B U , Z ( p ) ) H^{\ast }(BU,{{\mathbf {Z}}_{(p)}}) and H ∗ ( B U , Z ( p ) ) {H_{\ast }}(BU,{{\mathbf {Z}}_{(p)}}) where p p local lifts are determined by the values on primitives and Cartan formulas. This algorithm also provides a description of Steenrod and Dyer-Lashof operations on the fiber of any H H map (or infinite loop map) B U → B U BU \to BU , and applications to the classifying spaces of surgery which arise in this fashion will appear shortly.