A synthetic approach to detecting v₁-periodic families

2024/01/29 by Carrick, Christian, Davies, Jack Morgan · 2 citations
#55N20 #55P42 #55T05 #55T15 #55T25 #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2401.16508

Abstract

We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-J spectrum j is surjective on homotopy groups. This is achieved using a novel t-structure on the category of E-synthetic spectra and a specific construction of F_p- and BP-synthetic lifts of j. These synthetic lifts then easily produce modified Adams and Adams--Novikov spectral sequences for j which we use the prove the above detection statement, all without ever calculating F_p- or BP-homology nor the associated Ext groups.

Cited by

• Revisiting the β₁-action on the 3-primary stable homotopy groups of spheres
• Slice spectral sequences through synthetic spectra

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