Explicit construction of nontrivial elements for homotopy groups of classical Lie groups

Abstract

Nontrivial elements of homotopy groups for unitary, orthogonal, and symplectic groups are given explicitly. In particular, (a) representatives of generators of nontrivial homotopy groups of stable special unitary, orthogonal, and symplectic groups are constructed using Clifford algebras; (b) the values for ‘‘winding numbers’’ for stable SU, SO, and Sp are calculated for generators of homotopy groups; and (c) representatives of generators of homotopy groups Πn−2(O(n−1)), Π2n−2(U(n−1)), Π4n−2(Sp(n−1)) are given.