TOPOLOGY and GROUPOIDS

Ronald Brown

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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  xxvi+512 pages

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.

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 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.

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;
• the Phragmen-Brouwer property, proved using the groupoid van Kampen theorem, and applied to the Jordan Curve theorem;
• the equivalence between covering maps of spaces and covering morphisms of groupoids;
• the fundamental groupoid of an orbit space by a discontinuous action of a group as the orbit groupoid of the fundamental groupoid.

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 .

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
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. Bibliography, notes and other discussions are intended to put the work in context.

An e-version is available from the author: this has some colour and uses hyperref,
with full internal links, which many will find convenient for study

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'.

Reviews of previous editions.

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

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September 27, 2006