21. These well understood elements account for most elements of the stable homotopy groups of spheres in small dimensions. 22. Stephen Smale expressed the regular homotopy classes of immersions as the homotopy groups of a certain Stiefel manifold. 23. They can be used to construct the Greek letter infinite families in the stable homotopy groups of spheres. 24. Hurewicz theorem ), then the nonabelian cohomology may be thought of as a dual of homotopy groups . 25. Topological spaces with differing homotopy groups are never equivalent ( homeomorphic ), but the converse is not true. 26. These are related to relative homotopy groups and to " n "-adic homotopy groups respectively. 27. These are related to relative homotopy groups and to " n "-adic homotopy groups respectively. 28. Like the fundamental group or the higher homotopy groups of a space, homology groups are important topological invariants. 29. One of the most important problems in stable homotopy theory is the computation of stable homotopy groups of spheres. 30. In some cases it can be shown that the higher homotopy groups of " Y " are trivial.
