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

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 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 . We prove that is isomorphic to , where *V* is a topological vector space. This is used to prove that every element of 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 for , 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