On the Last Kervaire Invariant Problem

TL;DR

This paper proves the existence of smooth framed manifolds with Kervaire invariant one in dimension 126, completing the classification of such manifolds across all known dimensions.

Contribution

It establishes the existence of Kervaire invariant one manifolds in dimension 126, resolving the last open case of the Kervaire invariant problem.

Findings

Existence of Kervaire invariant one manifolds in dimension 126

Complete classification of dimensions with such manifolds

Confirmation that these manifolds only exist in specific dimensions

Abstract

We prove that the element $h_6^2$ is a permanent cycle in the Adams spectral sequence. As a result, we establish the existence of smooth framed manifolds with Kervaire invariant one in dimension 126, thereby resolving the final case of the Kervaire invariant problem. Combining this result with the theorems of Browder, Mahowald--Tangora, Barratt--Jones--Mahowald, and Hill--Hopkins--Ravenel, we conclude that smooth framed manifolds with Kervaire invariant one exist in and only in dimensions $2, 6, 14, 30, 62$, and $126$.