The authors recently proved that $\theta_j$ does not exist for $j > 6$. Here $\theta_j$ is a hypothetical element of order 2 in the stable homotopy groups of spheres in dimension $2^{j+1}-2$.

In 1960, Kevaire defined a $\mathbb{Z}/2$-valued invariant for closed, smooth manifolds with a stable framing. In geometric terms, the above result means that the only possible dimensions for such manifolds with nontrivial Kervaire invariant are

\[

2, 6,14, 30, 62, 126

\]

The first 5 dimensions were previously known to be realized, the first 3 by $S^j \times S^j$ for $j=1,2,3$. The status of $\theta_6$ (in dimension 126) remains open.

The theorem implies that the kernel and cokernel of the Kervair-Milnor map

\[

\Theta_n \to \pi_n^{st} / im(J)

\]are completely known finite abelian groups. Here $\Theta_n$ is the group of exotic smooth structures on $S^n$ and the map associates to it the underlying framed manifold. The image of $J:KO_{n+1} \to \pi_n^{st}$ realizes the different choices of framings on such homotopy spheres.

For further details see: http://www.math.rochester.edu/u/faculty/doug/kervaire.html

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