Algebraic topology Homotopy groups Problem

Kervaire invariant problem

Builds on Kervaire invariant Stable homotopy groups of spheres

Problem

For which integers $n = 4k + 2$ is there a closed framed smooth manifold of dimension $n$ whose Kervaire invariant equals $1$ ? Equivalently, in which degrees $n$ do the stable homotopy groups of spheres $\pi^s_n$ contain an element of Kervaire invariant $1$ ?

Remarks

Kervaire showed in 1960 that the invariant vanishes in dimension $10$ . Browder showed in 1969 that the invariant can be nonzero only in dimensions $n = 2^j - 2$ . Framed manifolds of Kervaire invariant $1$ exist in dimensions $2$ , $6$ , $14$ , $30$ and $62$ . Hill, Hopkins and Ravenel showed that none exist in dimensions $2^j - 2$ with $j \ge 8$ , that is, in dimension $254$ and above. This left dimension $126$ as the only open case; in 2024 Weinan Lin, Guozhen Wang and Zhouli Xu announced that a manifold of Kervaire invariant $1$ exists in dimension $126$ .

Reading path

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Kervaire invariant concept
Stable homotopy groups of spheres concept
Kervaire invariant problem problem

Tag	`Z95Q7B022R`
URI	`m2://Z95Q7B022R`
Kind	problem
Writing	W0 stub
MSC	55Q45, 57R15
Status	This copy of the site has no build data. Run `m2 status Z95Q7B022R`.
About	Kervaire invariant
Uses	Kervaire invariant, Stable homotopy groups of spheres
Changed	2 November 2026

Kervaire invariant problem

Problem

Remarks

Reading path

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