On the non-existence of elements of Kervaire invariant one

Douglas C. Ravenel; Michael A. Hill; Michael J. Hopkins

On the non-existence of elements of Kervaire invariant one

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arxiv 0908.3724 v4 pith:QWTOBMPC submitted 2009-08-26 math.AT math.GT

Michael A. Hill , Michael J. Hopkins , Douglas C. Ravenel This is my paper · ORCID

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keywords invariantkervairedimensionselementsexistonlyalgebraicbrowder

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We show that Kervaire invariant one elements in the homotopy groups of spheres exist only in dimensions at most 126. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. With the exception of dimension 126 this resolves a longstanding problem in algebraic topology.

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Towards the $p=3$ Kervaire Invariant Problem: The $E_2$-page for the homotopy fixed points spectral sequence computing $\pi_*({E_6}^{hC_9})$
math.AT 2025-07 conditional novelty 6.0 of 10

Under an unproved conjectural model for the C9-action on the Lubin-Tate spectrum E6, the authors compute the E2 page of the homotopy fixed points spectral sequence and prove an E6 detection theorem for the p=3 Kervair...

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