Maps between topological groups that are homotopic to homomorphisms

Abstract:

Let G be a compact connected group and let H be a locally compact abelian group. Denote by ${C_e}(G,H)$ 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 ${C_e}(G,H)$. We prove that ${C_e}(G,H)$ is isomorphic to $V \times {\text {Hom}}(G,H)$, where V is a topological vector space. This is used to prove that every element of ${C_e}(G,H)$ 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 ${\pi _n}(H) = 0$ for $n \geqq 2$, and that a compact connected abelian group admits essentially only one H-space structure.