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The Structure of Compact Groups
[book]

Karl H. Hofmann, Sidney A. Morris

2013
unpublished

As this is the only 1990s' English language treatise known to the reviewer which covers the structure theory of compact groups, no reference is made in this review to other earlier specific works. It is assumed that interested readers will by now be familiar with those earlier works. In the present treatise almost all of the classical areas are covered eg representation theory, duality theory. One exception being the work leading to Dynkin diagrams which the authors felt to be well and truly
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... e well and truly covered elsewhere. Also, the work devoted to profinite groups (projective limits of finite groups) is restricted to that needed for other parts, since the detailed investigation of such groups was too specialised away from the main thrust of the book. Indeed, the statement is made at one point that if the book were mainly about one thing it would be connected compact groups. Various aspects of cohomology are also included.(See later remarks about compact semigroups.) A notable feature of the book is that the development eschews differential topology and geometry on manifolds in favour of extensive use of the exponential function on a Banach Algebra. Of course this then involves a deal of category theory and homology theory in a number of places. Early on it is established that a compact Lie group is a compact group with no small subgroups. The development in many ways goes beyond that strictly needed just for compact groups. Often the approach is to develop in a general way then obtain the needed results for compact groups as specialisations. For example, the theory of linear Lie groups is extensively developed first. It is then proved that compact Lie groups are all linear Lie groups in which quotients are again compact Lie groups (hence linear again). The needed theorems for compact Lie groups are then derived from the linear Lie group theory. Although, of course, extensive use is made of projective limits; the authors repeatedly emphasise that some of the most delicate aspects of compact group structure need to be obtained other than by such methods. They then carry out the promised other approaches in later chapters. There is strong didactic emphasis on the need to settle the abelian cases then get at the general cases by using maximal (pro-) tori. The authors develop and make use of interesting facts (which they say are not widely reported) about the closure of commutator subgroups in compact Lie groups. Several references to Bourbaki are made, and it is stated that in many places Bourbaki has been influential. Avoidance of Stoke's Theorem on manifolds is cited as one important example. The final Chapters 11 and 12 reflect the authors particular research interests and involve extensive work on free compact groups on the one hand and consideration of certain cardinal invariants for topological groups on the

doi:10.1515/9783110296792
fatcat:5fsqd4psnfba3nzosroe5ocnbe