Topology of Lie Groups, I and II (Translations of Mathematical Monographs, Vol 91) by Mamoru Mimura and Hirosi Toda (PDF)

Description

Lie groups are very general mathematical objects that appear in numerous areas such as topology, functional analysis, and algebra, as well as differential geometry and differential topology. The purpose of these two parts is to provide a guide to the topology of Lie groups and homogeneous spaces by bringing together a wide range of results relating to them. The first part thoroughly studies topological properties of the classical groups as typical examples of Lie groups. In the second part, the authors study general properties of compact Lie groups, particularly the exceptional groups.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I just wanted to reply to the negative review (Encyclopedic, but not in a good way). This is not a textbook for learning algebraic topology. Instead, it’s a reference book for workers in the field. To paraphrase the other review, the value of this book is that it collects in one place many useful and interesting facts about Lie groups, which is its purpose.

⭐Highly recommend this to all students and researchers in the field of lie groups.

⭐Resource

⭐This translation of a Japanese 2 volume set on the topology of Lie groups covers many topics in algebraic topology, such as homology theory and stable homotopy, and geometry, such as vector bundles, and uses them to calculate the topological properties of Lie groups. While it defines many elementary concepts, such as that of a Lie group or a vector bundle, it barely explains anything. Theorems are stated, but rarely proved; instead, the authors in the foreward claim that the reader should be able to supply the proofs, as if the readers could develop the entire machinery of algebraic topology on their own. Occasionally results are proved, but only easy ones, as if the authors are just being lazy. Definitely no one could actually learn any topology from this book, but rather than just admitting this and assuming that the reader has the background, many basic concepts are defined and elementary theorems are stated, which just serves to waste space. The only real value of this book is that it collects in one place many useful and interesting facts about Lie groups, which is why I said it was encyclopedic. One could use it as a reference to look up the stable homotopy groups of classical Lie groups, for example, but not much more than that.One unusual feature of this book is that it also covers exceptional Lie groups, such as F4 or E8, which is pretty rare – offhand I can’t think of any other book that gives algebraic topological facts on them.

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