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Topology and Groupoids Paperback – February 24, 2006

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Editorial Reviews

Review

Brown's first five chapters are true topological background material,
stretching as they do from the topology of the real line to a discussion
of projective ("and other") spaces. It is only in the sixth chapter (p.
201 ff.) that we get to the all-important fundamental groupoid, but
thereafter things get off the ground very swiftly: homotopy theory,
cofibrations, computing fundamental groupoids (Van Kampen, the Jordan
Curve Theorem revisited), covering spaces, orbit spaces, and, indeed,
orbit groupoids. A broad palette.
I do believe in the general efficacy of the general categorical approach
in mathematics ...., and I find Brown's philosophy both attractive and
convincing. To wit (p. xx):
"In mathematics, and in many areas, analogies are not between objects
themselves, but between the relations between these objects. We will
define many constructions by their relations to all other objects of the
same type - this is called a 'universal property' ... All this is the
essence of the 'categorical approach', ... a major unifying force in the
mathematics of the twentieth century."
Two final observations. The back cover of Topology and Groupoids
displays a Venn diagram suggesting that, to borrow another word from
Grothendieck, the yoga of groupoids should be amenable eventually to
include, or engulf, such objects as groups, group actions, bundles of
groups (!), and even sets and equivalence relations. This in itself is a
very exciting prospect, alone worth the price of admission. ....... The
book is well written, indeed it is really a monograph composed by an
insider and an expert; it is very serious mathematics presented in a
sound pedagogical style: it is a very readable book equipped with fine
examples and many exercises; and its impact should be felt beyond the
confines of topology, even as topologists should be attracted to this
material most strongly.

Topology and Groupoids is an impressive work which should be given a
wide circulation.

Review by Michael Berg for the Mathematical Association of America

About the Author

Studied topology with JHC Whitehead and MG Barratt. Lectured at Liverpool, Hull and Professor at Bangor since 1970. Over 170 publications, (124 on MathSciNet, with 49 co-authors) on topology, algebra, category theory, and 36 on popularisation and teaching. Originator of the terms "convenient category" and "higher dimensional algebra". His latest publication is R. Brown, P.J. Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages. (August 2011) which gives an exposition of aspects of 40 years of research on developing applications of higher groupoids in topology.
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Product Details

  • Paperback: 538 pages
  • Publisher: BookSurge Publishing (February 24, 2006)
  • Language: English
  • ISBN-10: 1419627228
  • ISBN-13: 978-1419627224
  • Product Dimensions: 6 x 1.2 x 9 inches
  • Shipping Weight: 2 pounds (View shipping rates and policies)
  • Average Customer Review: 5.0 out of 5 stars  See all reviews (2 customer reviews)
  • Amazon Best Sellers Rank: #1,475,374 in Books (See Top 100 in Books)

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Top Customer Reviews

Format: Paperback
This book presents an unusual but very valuable approach to topology and homotopy in terms of groupoids, in particular, in terms of the fundamental groupoid. It is revised, and this is its third edition. It is self-contained and is beautifully written. The importance of groupoids have been amply justified by their increasing importance in noncommutative geometry and differential geometry. The book, for example, gives insight into the surprising abelian character of the higher homotopy groups by pointing out that we should be looking at the higher homotopy GROUPOIDS which are non-abelian.

The book is incredible value at $23.99. The print is excellent and delivery from BookSurge was in good time. There is also an e-version of the book. This book is wonderful value!
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Format: Paperback
This is an introductory book on topology, with a focus on homotopy theory. Is is written from a largely category-theoretical viewpoint. However, unlike other such treatments, such as May's "Concise course", it always keeps geometric intuition close at hand. This makes the category theory come very naturally, and also makes the book easier to understand for the non-category theorist. Since it also has very few prerequisites (e.g. no previous knowledge of topology or category theory is required), it would work very well for a first undergraduate course on topology. However, because of its insightful and slightly uncommon take on topology as a whole, I would guess that even some working mathematicians might learn a thing or two, or find new things to think about. The book as a whole indicates very well how a groupoidal way of doing one-dimensional homotopy theory is far more natural than the standard group-theoretical one. Now all we need is an equally natural and accessible extension to the higher homotopy groups.
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