ABSTRACT
There is a very concrete motivation for this section. Earlier in this volume† we constructed homomorphisms φ: SL(2, ℂ) → SO(3, 1) and T : S → O ( 3 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq3161.tif"/> for which we could determine the kernel, but not the image. A naive attempt to do so in a straightforward manner will lead to intractable calculations (as the reader is invited to find out for himself). New techniques are needed. It is a key observation that all the groups in our examples, namely SL(2, ℂ), SO(3, 1), S 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136547/a57ae1f0-6f59-46bd-bbdd-35a521b486a1/content/eq3162.tif"/> and O(3), are not just groups, but sit inside Euclidean spaces and inherit a topological and even an analytical structure from these surrounding spaces; they are analytic manifolds. Moreover, the homomorphisms φ and T respect this additional structure; they are continuous, even analytical mappings. If a given group G is (or can be) equipped with an additional structure like a topology or a manifold structure, we want to use this additional structure to learn more about the structure of G.
