A synthetic approach to detecting $v_1$-periodic families

Abstract

We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-$J$ spectrum $\mathrm{j}$ 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 ${\mathbf{F}}_p$- and BP-synthetic lifts of $\mathrm{j}$. These synthetic lifts then easily produce modified Adams and Adams--Novikov spectral sequences for $\mathrm{j}$ which we use the prove the above detection statement, all without ever calculating ${\mathbf{F}}_p$- or BP-homology nor the associated Ext groups.

53 pages and 16 spectral sequence diagrams. Comments are always welcome. v2 adds a proof of Wood's theorem and refines assumptions. Accepted version

References in corpus (4)

• The Adams spectral sequence for the image-of-$J$ spectrum
• Hecke operators on topological modular forms
• Uniqueness of real ring spectra up to higher homotopy
• Constructing and calculating Adams operations on dualisable topological modular forms