A note on the Greek letter elements of the homotopy of the L_n-local spheres

2025/07/02 by Ryo Kato, Kato, Ryo, Katsumi Shimomura +3
Mathematics · #55Q45 (Primary) 55Q51 (Secondary) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2507.02155

openalex publication_date 2025/07/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let BP denote the Brown-Peterson spectrum at a prime p, whose homotopy groups are isomorphic to the polynomial algebra generated by elements v_i's for i≥ 1. We consider the homotopy groups of the v_n^-1BP-localized sphere spectrum L_nS⁰ with n²≤ 2p-1, and show that the groups contain the n-th Greek letter family. For the proof of this, we further show the existence of the v_n^-1BP-localized Smith-Toda spectrum W_n for the case n²≤ 2p-1. If the Smith-Toda spectrum V(n-1) exists, then L_nV(n-1) is an example of W_n. Previously, W_n is shown to exist if n²+n≤ 2p-1. We also consider the case where (p,n)=(7,4), and show the existence of the delta family in π_*(L₄S⁰).

Citations

• Products of Greek letter elements dug up from the third Morava stabilizer algebra

Related

• Uniqueness of p-local truncated Brown-Peterson spectra
• Generalized Moore spectra and Hopkins' Picard groups for a smaller chromatic level
• A remark on Hopkins' chromatic splitting conjecture
• The homotopy groups of the E(2) local sphere at p > 3 revisited
• The Method of Infinite Descent in Stable Homotopy Theory II