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Let G be a compact connected group and let H be a locally compact abelian group. Denote by CC(G, H) the space of all identity preserving continuous functions from G to H with the compact-open topology, and denote by Hom(G, H) the space of all homomorphisms in Ce(G, H). We prove that Ce(G, H) is isomorphic to Vx Hom(G, H), where Vis a topological vector space. This is used to prove that every element of Ce(G, H) is homotopic to precisely one element of Hom(G, H). We also prove that the fundamental group of H is isomorphic to Hom(K, H), K being the circle group, that 7r(H)=O for n>2, and that a compact connected abelian group admits essentially only one H-space structure. Introduction. Plunkett [6] showed that every continuous function from a compact, connected, abelian, and 2? countable group into the circle group is homotopic to exactly one continuous homomorphism. Corollary 2 is a generalization of this. Enochs [3] proved Corollary 3 in the case where the group is compact. Notation. The phrase "continous function" will be shortened to "map", and "space" will mean "topological space". If X and Y are spaces, then C(X, Y) denotes the space of all maps from X to Y with the compactopen topology. If xeX then 7rv(X, x) is the nth homotopy group of X with base point x. Topological groups will always be assumed Hausdorff, and abelian groups will be written additively. If A and B are topological groups, then Ce(A, B) is {feC(A, B)If(1)=0}, where 1 and 0 are the respective identities, and Hom(A, B) is {fECe(A, B)If is a homomorphism}. One can verify that, if A is locally compact and B is abelian, then Ce(A, B) and Hom(A, B) are topological groups with the relative topology and pointwise operations. The group of real members and the group of integers are denoted by R and Z, respectively. We set K=R/Z, and p: R-*K is the natural map. Received by the editors June 18, 1971. AMS 1970 subject classifications. Primary 22C05, 54F60, 55A05, 55E05; Secondary 55D45.

Maps between topological groups that are homotopic to homomorphisms