Vol. 100, No. 1, 1982

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Real homology of Lie group homomorphisms

Robert F. Brown

Vol. 100 (1982), No. 1, 33–42

Let h : G1 G2 be a homomorphism of compact, connected Lie groups and let h : H(G1) H(G2) be the homomorphism of homology with real coefficients induced by h. The investigation of the properties of h that can be deduced from a knowledge of h goes back at least to work of Dynkin in the early 1950’s. This paper presents several contributions to the investigation. The main result is a characterization of homomorphisms with abelian images as those whose induced homomorphisms annihilate all three-dimensional indecomposables. We then examine what the homology can tell us about the dimension of the abelian image. Next, an inequality relating the homology of the kernel of h to the kernel of h leads to sufficient conditions for h to have an abelian, semisimple, or finite kernel. The final sections present various relationships between h and the kernel and image of h and, in particular, show that if h(G1) is totally nonhomologous to zero in G2, then h gives quite precise information about the behavior of h.

Mathematical Subject Classification 2000
Primary: 57T10
Secondary: 22E15
Received: 3 September 1980
Published: 1 May 1982
Robert F. Brown
Department of Mathematics
University of California,Los Angeles
Los Angeles CA 90095-1555
United States