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.
