2023/06/25 by Nicholas J. Kuhn, Kuhn, Nicholas J.
Mathematics · #55S12 (Primary) 55S10 #55T15 (Secondary) #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2306.14158
openalex publication_date 2023/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ExtAs,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 C2-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.