How Big are the Stable Homotopy Groups of Spheres?

Abstract

In this article we show that the $p$-torsion exponent of the stable stems grows sublinearly in $n$ and the $p$-rank of the $E_2$-page of the Adams spectral sequence grows as $\exp(\Theta( \log(n)^3))$. 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 $E_2$-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.