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Maps between topological groups that are homotopic to homomorphisms


Author: Wladimiro Scheffer
Journal: Proc. Amer. Math. Soc. 33 (1972), 562-567
MSC: Primary 22A05; Secondary 57E99
DOI: https://doi.org/10.1090/S0002-9939-1972-0301130-8
MathSciNet review: 0301130
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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.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0301130-8
Keywords: Compact-open topology, homotopy group, homotopic maps, H-space, normalized Haar integral, topological vector space
Article copyright: © Copyright 1972 American Mathematical Society

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