Topology and Groupoids by Ronald Brown (PDF)

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This is the third edition of a classic text, previously published in 1968, 1988, and now extended, revised, retitled, updated, and reasonably priced. Throughout it gives motivation and context for theorems and definitions. Thus the definition of a topology is first related to the example of the real line; it is then given in terms of the intuitive notion of neighbourhoods, and then shown to be equivalent to the elegant but spare definition in terms of open sets. Many constructions of topologies are shown to be necessitated by the desire to construct continuous functions, either from or into a space. This is in the modern categorical spirit, and often leads to clearer and simpler proofs. There is a full treatment of finite cell complexes, with the cell decompositions given of projective spaces, in the real, complex and quaternionic cases. This is based on an exposition of identification spaces and adjunction spaces. The exposition of general topology ends with a description of the topology for function spaces, using the modern treatment of the test-open topology, from compact Hausdorff spaces, and so a description of a convenient category of spaces (a term due to the author) in the non Hausdorff case. The second half of the book demonstrates how the use of groupoids rather than just groups gives in 1-dimensional homotopy theory more powerful theorems with simpler proofs. Some of the proofs of results on the fundamental groupoid would be difficult to envisage except in the form given: `We verify the required universal property’. This is in the modern categorical spirit. Chapter 6 contains the development of the fundamental groupoid on a set of base points, including the background in category theory. The proof of the van Kampen Theorem in this general form resolves a failure of traditional treatments, in giving a direct computation of the fundamental group of the circle, as well as more complicated examples. Chapter 7 uses the notion of cofibration to develop the notion of operations of the fundamental groupoid on certain sets of homotopy classes. This allows for an important theorem on gluing homotopy equivalences by a method which gives control of the homotopies involved. This theorem first appeared in the 1968 edition. Also given is the family of exact sequences arising from a fibration of groupoids. The development of Combinatorial Groupoid Theory in Chapter 8 allows for unified treatments of free groups, free products of groups, and HNN-extensions, in terms of pushouts of groupoids, and well models the topology of gluing spaces together. These methods lead in Chapter 9 to results on the Phragmen-Brouwer Property, with a Corollary that the complement of any arc in an n-sphere is connected, and then to a proof of the Jordan Curve Theorem. Chapter 10 on covering spaces is again fully in the base point free spirit; it proves the natural theorem that for suitable spaces X, the category of covering spaces of X is equivalent to the category of covering morphisms of the fundamental groupoid of X. This approach gives a convenient way of obtaining covering maps from covering morphisms, and leads easily to traditional results using operations of the fundamental group. Results on pullbacks of coverings are proved using a Mayer-Vietoris type sequence. No other text treats the whole theory directly in this way. Chapter 11 is on Orbit Spaces and Orbit Groupoids, and gives conditions for the fundamental groupoid of the orbit space to be the orbit groupoid of the fundamental groupoid. No other topology text treats this important area. Comments on the relations to the literature are given in Notes at the end of each Chapter. There are over 500 exercises, 114 figures, numerous diagrams. See http://www.bangor.ac.uk/r.brown/topgpds.html for more information. See http://mathdl.maa.org/mathDL/19/?rpa=reviews&sa=viewBook& bookId=69421 for a Mathematical Association of America review.

User’s Reviews

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.

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

⭐More usual or standard introductions to topology strain to include topics in service to analysis. That decreases the time it takes to expose curriculum to students, but it does so at the expense of real understanding. This is a very fine text. It will let you study topology ab initio on topological terms and not substantially as a service to other areas. It is a little unconventional, but for most excellent reasons. This text does it right. Others are simply not wrong. Use this one, not Kelley, not Dugundji. Bourbaki is always worth reading. Follow this with Bott and Tu.

⭐I was a teenager when I first read–or struggled with–“Topology and Groupoids” in its first edition, “Modern Topology.” Later in college, a differential topologist told me that Modern Topology was idiosyncratic for its emphasis on the fundamental groupoid, and it took too long to reach covering spaces. He couldn’t recommend it. His preference was for Singer and Thorpe’s more traditional text. Now with experience (and CPAP therapy, known to lead to significant cognitive improvement in patients with severe sleep apnea), “Topology and Groupoids” remains my favorite text on topology for its vision, far ahead of its time; its emphasis on homotopy; and for its efficient proofs. If only I had attained something of my current appreciation when I was a shy, self-conscious teenager.

⭐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.

⭐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!

⭐I do not understand why I never see recommendations for this book. I come about this book while finding a reference for fundamental groupoid. A good book, contains at least nice, succinct and complete proofs. But Brown brings this to another level, with historical references in the end, time to time, he analyzes the meaning of each definition, lemma, theorem and corollary. This is evident even in small chapters as 5.9. I have learnt a LOT from these insights.I believe this is in spirt of how we learn mathematics – as I would quote him from the book, “…often the important question is not: What is the answer? but instead is: What is the question?” What is most captivating is the categorical viewpoint. Brown let us understand both global and local aspects of topology. In comparison in some texts where the “intuition” given are often well dressed stories that are ad hoc.In comparison to other texts, as Hatcher’s, Rotman’s, Bredon’s, or Massey’s, this book is just NEATER. Unfortunately, the author has not written anything about homology theory. A great pity for such a mathematician and teacher.

⭐This is my favourite introductory topology book around. Professor Brown freely incorporates categorical terminology and concepts, which gives a sense of clarity and purpose to the reader. It shows how constructions like quotients, products, and coproducts fit into a more general categorical framework that transcends the field of topology. But best of all, the text focuses on the more general construction of a fundamental groupoid, as opposed to fundamental groups, which require the choice of a base point.

⭐Lo consiglio a tutti coloro che apprezzano la teoria delle categorie e che voglio fare topologia in un modo innovativo e più generale.Il testo è chiarissimo e scorrevole.original et exemples tirés de la recherche en géométrie, intuitif mais rigoureux; certains développements actuels sont toutefois manquants…groupoïdes vu de manière originale

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