Fake Lie groups and maximal tori. II (Q911093)

[Part I is reviewed above (see Zbl 0697.55006).] A finite loop space X is called a fake Lie group of type G, where G is a compact connected Lie group, if the classifying spaces BX and BG have the same genus. (Recall that two nilpotent spaces of finite type \(Y'\), \(Y''\) are said to have the same genus if their p-localizations are homotopy equivalent for all primes.) Rector's maximal torus theorem may be interpreted as using the existence of a maximal torus for \({\mathbb{S}}p(1)\) to distinguish \({\mathbb{S}}p(1)\) from a fake \({\mathbb{S}}p(1)\). In this note we prove the analog of Rector's theorem for \({\mathbb{U}}(2)\). As a bonus we are able to characterize B\({\mathbb{U}}(2)\) up to homotopy type by its K-theory.