that bound a parallelizable manifold, is the th stable homotopy group of spheres, and is the image of the -homomorphism. This is an isomorphism unless is of the form , in which case the image has index 1 or 2. The groups above, and therefore the stable homotopy groups of spheres, are used in the classification of possible smooth structures on a topological or piecewise linear manifold. The Kervaire invariant problem, about the existence of manifolds of Kervaire invariant 1 in dimensions can be reduced to a question about stable homotopy groups of spheres. For example, knowledge of stable homotopy groups of degree up to 48 has been used to settle the Kervaire invariant problem in dimension . (This was the smallest value of for which the question was open at the time.) The Barratt–Priddy theorem says that the stable homotopy groups of the spheres can be expressed in terms of the plus construction applied to the classifying space of the symmetric group, leading to an identification of K-theory of the field with one element with stable homotopy groups. Table of homotopy groups Tables of homotopy groups of spheres are most conveniently organized by showing . The following table shows many of the groups . The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following conventions: The entry "⋅" denotes the trivial group. Where the entry is an integer, , the homotopy group is the cyclic group of that order (generally written ). Where the entry is ∞, the homotopy group is the infinite cyclic group, . Where entry is a product, the homotopy group is the cartesian product (equivalently, direct sum) of the cyclic groups of those orders. Powers indicate repeated products. (Note that when and have no common factor, is isomorphic to .) Example: , which is denoted by in the table. Table of stable homotopy groups The stable homotopy groups are the products of cyclic groups of the infinite or prime power orders shown in the table. (For largely historical reasons, stable homotopy groups are usually given as products of cyclic groups of prime power order, while tables of unstable homotopy groups often give them as products of the smallest number of cyclic groups.) The main complexity is in the 2-, 3-, and 5-components: