Kervaire invariant one problem

Solved, 2009Topology

Statement

Determine all dimensions $n = 4k + 2$ in which there exists a smooth, closed, stably framed manifold $M^n$ whose Kervaire invariant $\Phi(M^n) \in \mathbb{Z}/2\mathbb{Z}$ (the Arf invariant of the quadratic refinement of the middle-dimensional intersection form on $H_{2k+1}(M^n; \mathbb{Z}/2\mathbb{Z})$ induced by the stable framing) is equal to $1$ .

In 1969 William Browder proved that a smooth closed framed manifold of Kervaire invariant $1$ can exist only in dimensions $n = 2^{j+1} - 2$ , and only if the class $h_j^2$ in the $E_2$ -page of the $\bmod\,2$ Adams spectral sequence survives to represent an element $\theta_j$ in the stable homotopy group of spheres $\pi_{2^{j+1}-2}^S$ . Constructions in dimensions $2, 6, 14, 30$ , and $62$ ( $1 \le j \le 5$ ) were completed by 1984 (culminating in work of Michael Barratt, John Jones, and Mark Mahowald). In 2009 (published in Annals of Mathematics in 2016), Michael A. Hill, Michael J. Hopkins, and Douglas C. Ravenel proved using equivariant stable homotopy theory and the $C_8$ slice spectral sequence that $\theta_j$ does not exist for any $j \ge 7$ (dimensions $254$ and above), leaving only dimension $126$ ( $j = 6$ ) open. In December 2024, Weinan Lin, Guozhen Wang, and Zhouli Xu posted a preprint (`arXiv:2412.10879`, not yet peer-reviewed) proving via machine-assisted Adams spectral sequence computations that $h_6^2$ is a permanent cycle, establishing existence in dimension $126$ .

The Kervaire invariant problem sits at the heart of the classification of exotic smooth spheres initiated by Michel Kervaire and John Milnor in 1963: the group $\Theta_n$ of homotopy $n$ -spheres is closely controlled by the stable homotopy groups of spheres $\pi_n^S$ and the Kervaire invariant homomorphism. The equivariant homotopy machinery introduced by Hill, Hopkins, and Ravenel—specifically the Hill–Hopkins–Ravenel norm functor and the slice filtration for $G$ -spectra—transformed equivariant and chromatic homotopy theory far beyond the Kervaire problem.

References

William Browder (1969). The Kervaire invariant of framed manifolds and its generalization · DOI:10.2307/1970747
Michael A. Hill, Michael J. Hopkins, Douglas C. Ravenel (2016). On the nonexistence of elements of Kervaire invariant one · DOI:10.4007/annals.2016.184.1.1 · arXiv:0908.3724
Weinan Lin, Guozhen Wang, Zhouli Xu (2024). On the Last Kervaire Invariant Problem · arXiv:2412.10879 [preprint, not peer-reviewed]

History of the problem

1960Michel Kervaire introduces the $\mathbb{Z}/2\mathbb{Z}$ -valued Kervaire invariant of a $(4k+2)$ -dimensional framed manifold and uses it to construct a $10$ -dimensional piecewise-linear manifold that admits no smooth structure.
1969William Browder proves that smooth closed framed manifolds of Kervaire invariant one can exist only in dimensions $n = 2^{j+1} - 2$ , and only when $h_j^2$ survives in the $\bmod\,2$ Adams spectral sequence.
1984Michael Barratt, John Jones, and Mark Mahowald establish the existence of a Kervaire invariant one element $\theta_5$ in dimension $62$ , completing the positive cases $n = 2, 6, 14, 30, 62$ .
2009Michael Hill, Michael Hopkins, and Douglas Ravenel announce their proof that no smooth framed manifolds of Kervaire invariant one exist in dimensions $2^{j+1} - 2$ for any $j \ge 7$ (dimensions $254$ and higher), refuting the "Doomsday Conjecture" and leaving only dimension $126$ open.
2024Weinan Lin, Guozhen Wang, and Zhouli Xu post a preprint (`arXiv:2412.10879`) proving that $h_6^2$ is a permanent cycle in the Adams spectral sequence, establishing the existence of smooth framed manifolds of Kervaire invariant one in the final open dimension $126$ .