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Fundamental Groups and Covering Spaces

Elon Lages Lima, translated by Jonas Gomes
AK Peters
Publication Date: 
Number of Pages: 
[Reviewed by
Darren Glass
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AK Peters has recently published the first translation into English of Elon Lages Lima's book Fundamental Groups and Covering Spaces, which was originally published in Portuguese as part of IMPA's "Project Euclides" series. The book introduces the reader to some of the key ideas of algebraic topology, and it does so with clear, easy-to-read exposition and many examples. I found the book an enjoyable read, even knowing most of the punchlines ahead of time.

The book is divided into two parts. The first part is about homotopy, fundamental groups, and winding numbers. This reviewer found the chapter on the classical matrix groups, in which Lima looks at the groups SO(n), SU(n), and Sp(n) and computes what their fundamental groups are, to be especially interesting and novel to a book at this level. The second part of the book deals with covering spaces, starting with local homeomorphisms, liftings, and covering maps and working through a classification of the fundamental groups of compact surfaces. He concludes with a discussion of orientability in general, oriented double coverings in particular, and a nice discussion of the fact that the fundamental group of a nonorientable manifold has a subgroup of index two.

In addition to the lucid writing — for which translator Jonas Gomes certainly deserves a piece of the credit — and plentiful exercises, another nice feature of Lima's book is the number of references to the history of the ideas which he is presenting. While the book is by no means a history book, it does contain more references to the literature than one typically find in undergraduate textbook, which I found quite refreshing. All in all, I would not hesitate to recommend this book to an undergraduate wanting to learn the subject.

Darren Glass is a VIGRE Assistant Professor of Mathematics at Columbia University. His research interests include number theory, algebraic geometry, and cryptography. He can be reached at

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