Homotopy groups of compact Abelian groups

Abstract

Preliminaries. Bn (resp. Sn_1) will denote the subset of Rn consisting of those x such that IIx|| < 1 (resp. ||x|I = 1). xo will denote the point (1, 0, 0, * , 0) of Rn and T will denote S1 made into a topological group by using complex multiplication. All groups will be assumed to be Abelian. For a based topological space X and a topological group G, let ((X, G) denote the set of maps (i.e., base point preserving continuous maps) of X into G (where G is considered to be a based topological space with base point 0). In an obvious fashion, ((X, G) can be endowed with a group structure. In case X is also a topological group (considered as a based topological space with base point 0) let Hom(X, G) denote the subgroup of C(X, G) consisting of those maps which are homomorphisms. The spaces Bn and Sn-1 will all be assumed to be based with x0 as a base point. If G is a discrete or compact group we let G* denote the group Hom(G, T) endowed with the topology of compact convergence.If G is compact (resp. discrete) then it is known that G* is discrete (resp. compact). Also for two discrete or two compact groups G1 and G2 we have an isomorphism from the group Hom(Gi, G2) onto Hom(G*, G*) where an element fGHom(G1, G2) is mapped onto its transpose f*. Furthermore a sequence of compact (or discrete) groups