for , . The homology groups , with , are all trivial. It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general. This is the case that is of real importance: the higher homotopy groups , for , are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics. Table The following table gives an idea of the complexity of the higher homotopy groups even for spheres of dimension 8 or less. In this table, the entries are either the trivial group 0, the infinite cyclic group , finite cyclic groups of order (written as ), or direct products of such groups (written, for example, as or ). Extended tables of homotopy groups of spheres are given at the end of the article. The first two rows of this table are straightforward. The homotopy groups of the 0-dimensional sphere are trivial for , because any base point preserving map from an -sphere to a 0-sphere is a one-point mapping. Similarly, the homotopy groups of the 1-sphere are trivial for , because the universal covering space, , which has the same higher homotopy groups, is contractible. Beyond these two rows, the higher homotopy groups () appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle. The groups below the jagged black line are constant along the diagonals (as indicated by the red, green and blue coloring). Most of the groups are finite. The only infinite groups are either on the main diagonal or immediately above the jagged line (highlighted in yellow). The third and fourth rows of the table are the same starting in the third column (i.e., for ). This isomorphism is induced by the Hopf fibration . For and the homotopy groups do not vanish. However, for . These patterns follow from many different theoretical results. Stable and unstable groups The fact that the groups below the jagged line in the table above are constant along the diagonals is explained by the suspension theorem of Hans Freudenthal, which implies that the suspension homomorphism from to is an isomorphism for . The groups with are called the stable homotopy groups of spheres, and are denoted : they are finite abelian groups for , and have been computed in numerous cases, although the general pattern is