Dyer-Lashof operations as extensions of Brown-Gitler Modules

TL;DR

This paper constructs explicit Dyer-Lashof operations as extensions of Brown-Gitler modules at prime 2, linking algebraic and geometric perspectives to enhance understanding of their actions on spectra and Ext groups.

Contribution

It provides explicit constructions of Dyer-Lashof operation extensions related to cofiber sequences and spectra, connecting algebraic and geometric frameworks.

Findings

Extensions induce Dyer-Lashof algebra actions on Ext groups.

Cofibration sequences realize these extensions geometrically.

Spectral sequences of modules over the Dyer-Lashof algebra are established.

Abstract

At the prime 2, let T(n) be the n dual of the nth Brown-Gitler spectrum with mod 2 homology G(n). Our previous work on computing the homology of an infinite loopspaces led us to observe that there are extensions between various of the right A-modules G(n) such that splicing with these gives an action of the Dyer-Lashof algebra on the sum over s and n of Ext_A^{s,s}(G(n),M). We give explicit constructions of these `Dyer-Lashof operation' extensions: one construction relates them to the cofiber sequence associated to the C_2-transfer. Another relates key `squaring' Dyer-Lashof operations to the Mahowald short exact sequences. Finally, properties of the spectra T(n) allow us to geometrically realize our extensions by cofibration sequences, with the implication that the sum over n of all the Adams spectral sequences computing [T(n),X] is a spectral sequence of modules over the Dyer-Lashof algebra.