# TOPOLOGY and GROUPOIDS

## Ronald Brown

 ......people are accustomed to work with fundamental groups and generators and relations for these and stick to it, even in contexts when this is wholly inadequate, namely when you get a clear description by generators and relations only when working simultaneously with a whole bunch of base-points chosen with care - or equivalently working in the algebraic context of groupoids, rather than groups. Choosing paths for connecting the base points natural to the situation to one among them, and reducing the groupoid to a single  group, will then hopelessly destroy the structure and inner symmetries of the situation, and result in a mess of generators and relations no one dares to write down, because everyone feels they won't be of any use whatever, and just confuse the picture rather than clarify it. I have known such perplexity myself a long time ago, namely in Van Kampen type situations, whose only understandable formulation is in terms of (amalgamated sums of) groupoids. Alexander Grothendieck (for more of his comments on mathematics click here) many base points I= π1([0,1],{0,1}) is the groupoid with two objects 0,1 and 4 arrows (see p.xxi)

Link to a review for the Math. Assoc. America by Michael Berg, Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.
The Basic Library List Committee of the MAA suggests that undergraduate mathematics libraries consider this book for acquisition.

A very geometric approach to the fundamental groupoid can be found in Ronald Brown's Topology and Groupoids. Since EVERYTHING is expressed from the beginning in terms of the category of equivelence classes of paths,the formulation is very straightforward and simple. I highly recommend the book to all mathematicians: I have seen the future of point-set topology courses and Brown's text is the crystal ball.
Andrew L.
Jun 9 2010 at 7:03 Mathoverflow

The book is used for a course at Harvard.

For my Honours module on general topology, I am following your book "Topology and Groupoids". My students are enjoying it quite a lot since it has lots of motivations before introducing some new ideas, definitions and theorems.
Dr. Amartya Goswami, Department of Mathematical Sciences, University of Zululand.

November, 2012 "I was also very pleased to get a chance to read your book. Your exposition style is excellent, clear, concise, and precise."
Jose Ignacio Cogolludo Agustin, Area de Geometria y Topologia, IUMA, Departamento de Matematicas, Universidad de Zaragoza

"One of the nicest features of this book is the joy and enthusiasm that pervade it." From a review of the second edition by Charles A. McGibbon for MathSciNet.

Link to a review by Vagn Lundsgaard Hansen. (pdf) in the Bull. London Math. Soc.

Link to an online review (though the price quoted is \$23.99 not the current \$31.99).

My own groupoid web page

"Excellent perspective on topology" 5 out of 5 stars 11 Feb 2013
By Staffan Angere, Lund University - Published on Amazon.com
This is an introductory book on topology, with a focus on homotopy theory. It 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.

Here is a link to a review of groupoids in the context of stacks in geometry and physics.

From groups to groupoids: a brief survey  Bull LMS 19 (1987) 113-134.    Link to pdf file.

"May I take this opportunity to write that I am thoroughly enjoying your book. I am still at the very beginning but I really appreciate the way you explore the full implications of definitions."

Andrew Hull, December 2009

## Reviews of previous editions.

Order
ISBN: 1-4196-2722-8; Library of Congess Control Number: 2006901092
There are over 500 exercises, 114 figures, numerous diagrams.
Printed and Distributed by Booksurge LLC, March 2006.  Price: \$31.99  xxv+512 pages
Available through amazon.com (printed in USA)  or UK and Europe amazon sites (printed in these countries). Also Abe Books

Also available through EBSCOhost databases.

e-book £5 through Kagi https://store.kagi.com/cgi-bin/store.cgi?storeID=6FEPD_LIVE
(this version has full hyperref with full internal links, which many will find convenient for study, and some colour).
(Problem: Unarchive using Mac OS 10+. The standard Mac utility Archive does not succeed even before it gets to requesting any password. Solution: Download Zipeg from the Internet. It is Shareware and reported to work a treat.)

To get this book to you at these low prices has been accomplished through printing outside a conventional scientific publishing house. This also means all publicity was initiated by me.
If you like the book, please help by telling people, giving more recommendations on the amazon sites, and if possible arranging further reviews, or even translations.
What possibilities are there for public development of this material, based on the LaTeX files?

 A geometric account of general topology, homotopy types, and the fundamental groupoid

This is a retitled,  revised, updated and extended edition of a classic text, first published in 1968. (Sample Chapter as pdf file: 9 Computation of the fundamental groupoid).

Its first half gives a geometric account of general topology appropriate to a beginning course in algebraic topology. For example, it includes identification spaces, adjunction spaces and finite cell complexes, and a convenient category of spaces.

The second half introduces the algebra of groupoids and shows the utility of this algebra for modelling geometry. It is also one of the few basic topology texts to emphasise the importance of categorical methods and universal properties in allowing analogies between different mathematical structures. Thus the notion of pushout is used to describe (i) ways of constructing topological spaces from basic examples, and (ii) how a variety of modelling and of calculations follows from the way the fundamental groupoid on a set of base points preserves certain pushouts of spaces. 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'. An example is the main result on orbit spaces and orbit groupoids in Chapter 11 (this is published nowhere else).

The overall aim was to give the best context for the required theorems, so that they seem natural, simpler to prove and in the strongest form; in other words, to understand the mathematics!  The resulting exposition is non standard, and has been described as `idiosyncratic'! So be it!

Here are some examples of topics covered not available in other texts at this level:

• the initial topology on joins of spaces;
• the convenient category of k-spaces, in the non Hausdorff case;
• a gluing theorem for homotopy equivalences (this result has now become a standard lemma in abstract homotopy theory, but first appeared in the 1968 edition of this book; the proof here has the advantage of giving control of the homotopies involved);
• the notion of a path in a space X as a map from [0,r] to X, for r ≥ 0, gives a category of paths. This makes the work easier for students, since associativity comes for free, and also shows a purely algebraic use of categories: (this approach is also taken in the book on "Knot Theory" by Crowell and Fox);
• this book is the only topology text in English to prove a version, dating from 1967, of the van Kampen Theorem for the fundamental groupoid on a set of base points, and so deduce the fundamental group of the circle - for more discussion on many base points see this mathoverflow discussion ;
• the Jordan Curve theorem, proved via the Phragmen-Brouwer property, and as a consequence of the groupoid (many base point) van Kampen theorem;
• the equivalence between the categories of covering maps of spaces and covering morphisms of groupoids (this is surely the right base point free approach to covering space theory, since a covering map is modelled by a covering morphism, which makes it easier to deal with lifting problems for maps );
• the fundamental groupoid of an orbit space by a discontinuous action of a group as the orbit groupoid of the fundamental groupoid (this is a powerful result not generally recognised). Ross Geoghegan in his 1986 review (MR0760769) of two papers by M.A. Armstrong on the fundamental groups of orbit spaces wrote: "These two papers show which parts of elementary covering space theory carry over from the free to the nonfree case. This is the kind of basic material that ought to have been in standard textbooks on fundamental groups for the last fifty years." At present, to my knowledge, "Topology and Groupoids" is the only topology text to cover such results.

I hope that the availability of this text will help further investigation of related topics. For example, can these methods be applied to areas such as braid groups and mapping class groups? Also, this text should be a useful foundation for those wishing to study the applications of higher homotopy groupoids to Nonabelian algebraic topology . This is quite an open field with not much competition in algebraic topology, as the workers there hardly use groupoids at all. However groupoids are widely used in noncommutative geometry, certain areas of differential topology and the theory of stacks and gerbes.

List of chapters:

1. Some topology on the real line
2. Topological spaces
3. Connected spaces, compact spaces
4. Identification spaces and cell complexes
5. Projective and other spaces
6. The fundamental groupoid
7. Some combinatorial groupoid theory
8. Cofibrations
9. Computation of the fundamental groupoid [pdf]
10. Covering spaces, covering groupoids
11. Orbit spaces, orbit groupoids
12. Conclusion

Also included: Prefaces; Appendix on set theory and cardinality; glossary of terms from set theory; glossary of symbols. The bibliography, notes and other discussions are intended to put the work in context.

Here is a supplement to Chapter 6 with more on the van Kampen theorem for groupoids, and its applications. (September 14, 2011).

Licenses for the use of the e-version for class use may be negotiated with the author.

The author is indebted to Tony Bak and Peter May for their strong support of the proposal for a Leverhulme Fellowship, 2002-2004, for which work on this book formed the first part. He also thanks John Robinson and Ben Dickens for the cover design, derived from John Robinson's sculpture `Journey'.

ERRATA (Updated August 25, 2015)

Given as html file. This list gives all detailed errata onwards from the first printing. A further availability is of a bibliography, which is better ordered than the current one. This may be downloaded as pdf file here.

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