2022/03/01 by Robert Burklund, Andrew Senger, Burklund, Robert +1
Computer Science · Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2203.00670
openalex publication_date 2022/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we show that the p-torsion exponent of the stable stems grows sublinearly in n and the p-rank of the E2-page of the Adams spectral sequence grows as exp(Θ( log(n)3)). Together these bounds provide the first subexponential bound on the size of the stable stems. Conversely, we prove that a certain, precise, version of the failure of the telescope conjecture would imply that the upper bound provided by the Adams E2-page is essentially sharp -- answering the titular question: As big as the fate of the telescope conjecture demands. In an appendix joint with Andrew Senger we consider the unstable analog of this question. Bootstrapping from the stable bounds we prove that the size of the p-local homotopy groups of spheres grows like exp(O(log(n)3)), providing the first subexponential bound on the size of the unstable stems.