......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 basepoints 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.  
(for more of his comments on mathematics click here) 
I= π_{1}([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
pointset 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.
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.
Here is a 1 page advertisement/information leaflet.
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
5.0 out of 5 stars Excellent perspective on topology 11 Feb 2013
By Staffan Angere  Published on Amazon.com
This is an introductory book on topology, with a focus on homotopy theory. It is written from a
largely categorytheoretical 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 noncategory 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
onedimensional homotopy theory is far more natural than the standard grouptheoretical 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) 113134. 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
Order
ISBN: 1419627228; 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.
ebook
£5 through Kagi
https://store.kagi.com/cgibin/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

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:
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:
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 eversion 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, 20022004, 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 April 4, 2014)
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.
Return to Ronald Brown's home page
April 10, 2014