On the Last Kervaire Invariant Problem

TL;DR

This work resolves the final case of the Kervaire invariant problem by proving that the Adams spectral sequence element $h_6^2$ is a permanent cycle, thereby constructing framed manifolds with Kervaire invariant one in dimension $126$. The authors develop a comprehensive framework around $\mathrm{H}\mathbb{F}_2$-synthetic spectra, notably introducing the $f$-extension spectral sequence, the Generalized Leibniz Rule, and the Generalized Mahowald Trick to translate extension data into Adams differentials. Through a combination of synthetic Toda brackets, rigidity results, and careful computational analysis (aided by Lin’s software), they show that $h_6^2$ survives to $E_\infty$, completing the known spectrum of dimensions with Kervaire invariant one: $2,6,14,30,62,126$. The results solidify the link between synthetic-homotopical methods and classical stable homotopy theory, and they provide a detailed computational backbone for the asserted permanence of $h_6^2$, with implications for both manifold topology and unstable homotopy theory. The Appendix documents extensive Adams differential data used to support the arguments, reflecting a broad computational effort underlying the theoretical advances.

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 $126$.

Figures (9)

• Figure 1: A crossing of $d_n^f$ that hits Adams filtration $p$
• Figure 2: A demonstration of Theorem \ref{['thm:4114f70c']}
• Figure 3: A crossing of $d_r(x)=y$ on the $E_{n+1}$-page
• Figure 4: $(f,E_r)$-extension
• Figure 5: A crossing of an $(f,E_r)$-extension

Theorems & Definitions (119)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3: Browder Browder
• Theorem 1.4: Theorem \ref{['thm:126survives']}
• Corollary 1.5
• proof
• Definition 2.1
• Definition 2.3
• Proposition 2.5
• proof