for naming Bernoulli numbers; this article uses the number theorists' convention.) Map between quotients The quotient group 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 where is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism. As with , the image of J is a cyclic group, and is trivial or order 2 except in case , in which case it can be large, with its order related to the Bernoulli numbers. The quotient group is the "hard" part of the stable homotopy groups of spheres, and accordingly 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 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 remaining open; see that article for details. This is primarily the work of , which proved that such manifolds only existed in dimension , and , which proved that there were no such manifolds for dimension and above. Manifolds with Kervaire invariant 1 have been constructed in dimension 2, 6, 14, 30, and 62, but dimension 126 is open, with no manifold being either constructed or disproven. Order of Θn The order of the group is given in this table from (except that the entry for is wrong by a factor of 2 in their paper; see the correction in volume III p. 97 of Milnor's collected works). {| class="wikitable" style="text-align:center" |- ! Dim n !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! 17 !! 18 !! 19 !! 20 |- ! order | 1 || 1 || 1 || 1 || 1 || 1 || 28 || 2 || 8 || 6 || 992 || 1 || 3 || 2