Editing Exotic sphere (section)

===Map between quotients===
The quotient group <math>\Theta_n/bP_{n+1}</math> has a description in terms of [[stable homotopy groups of spheres]] modulo the image of the [[J-homomorphism]]; it is either equal to the quotient or index 2. More precisely there is an injective map
:<math>\Theta_n/bP_{n+1}\to \pi_n^S/J,</math>
where <math>\pi_n^S</math> is the ''n''th stable homotopy group of spheres, and ''J'' is the image of the ''J''-homomorphism. As with <math>bP_{n+1}</math>, the image of ''J'' is a cyclic group, and is trivial or order 2 except in case <math>n = 4k+3</math>, in which case it can be large, with its order related to the [[Bernoulli number]]s. The quotient group <math>\pi_n^S/J</math> is the "hard" part of the stable homotopy groups of spheres, and accordingly <math>\Theta_n/bP_{n+1}</math> is the hard part of the exotic spheres, but almost completely reduces to computing homotopy groups of spheres. The map is either an isomorphism (the image is the whole group), or an injective map with [[Index of a subgroup|index]] 2. The latter is the case if and only if there exists an ''n''-dimensional framed manifold with [[Kervaire invariant]] 1, which is known as the [[Kervaire invariant problem]]. Thus a factor of 2 in the classification of exotic spheres depends on the Kervaire invariant problem.

The Kervaire invariant problem is almost completely solved, with only the case <math>n=126</math> remaining open, although Zhouli Xu (in collaboration with Weinan Lin and Guozhen Wang), announced during a seminar at Princeton University, on May 30, 2024, that the final case of dimension 126 has been settled and that there exist manifolds of Kervaire invariant 1 in dimension 126.<ref>{{Cite web |title=Computing differentials in the Adams spectral sequence {{!}} Math |url=https://www.math.princeton.edu/events/computing-differentials-adams-spectral-sequence-2024-05-30t170000 |access-date=2025-05-04 |website=www.math.princeton.edu}}</ref> Previous work of {{harvtxt|Browder|1969}}, proved that such manifolds only existed in dimension <math>n=2^j-2</math>, and {{harvtxt|Hill|Hopkins|Ravenel|2016}}, which proved that there were no such manifolds for dimension <math>254=2^8-2</math> and above. Manifolds with Kervaire invariant 1 have been constructed in dimension 2, 6, 14, 30. While it is known that there are manifolds of Kervaire invariant 1 in dimension 62, no such manifold has yet been constructed. Similarly for dimension 126.