Nonvanishing of products in $v_2$-periodic families at the prime $3$

Christian Carrick; Jack Morgan Davies

Nonvanishing of products in v₂-periodic families at the prime 3

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arxiv 2410.02564 v3 pith:HACK4QF7 submitted 2024-10-03 math.AT math.KT

Nonvanishing of products in v₂-periodic families at the prime 3

Christian Carrick , Jack Morgan Davies This is my paper

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keywords familiesperiodicproductsactionadamsadams--novikovaffordsamongst

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Many products amongst $v_2$-periodic families in the stable homotopy groups of spheres are shown not to vanish and some Toda brackets are shown not to contain zero. This is done by carefully studying the action of Adams operations on topological modular forms. A crucial ingredient is Pstragowski's category of synthetic spectra which affords us the necessary freedom to work with (modified) Adams--Novikov spectral sequences.

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Revisiting the $\beta_1$-action on the $3$-primary stable homotopy groups of spheres
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Products of the 3-primary β₁-periodic family in the stable homotopy groups of spheres are nonzero for up to five factors and zero for six or more, with analogous sharp cutoffs for β₂-, α₁β₂-, and [α₁β₃/3]-twisted products.