Topology and groupoids 1st Edition by Ronald Brown ISBN 978-1419627224 1419627228
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ISBN 10: 1419627228
ISBN 13: 978-1419627224
Author: Ronald Brown
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.
Table of contents:
1 Some topology on the real line
1.1 Neighbourhoods in R.
1.2 Continuity
1.3 Open sets, closed sets, closure
1.4 Some generalisations
2 Topological spaces
2.1 Axioms for neighbourhoods
2.2 Open sets
2.3 Product spaces
2.4 Relative topologies and subspaces
2.5 Continuity
2.6 Other conditions for continuity
2.7 Comparison of topologies, homeomorphism
2.8 Metric spaces and normed vector spaces
2.9 Distance from a subset
2.10 Hausdorff spaces
3 Connected spaces, compact spaces
3.1 The sum of topological spaces
3.2 Connected spaces.
3.3 Components and locally connected spaces
3.4 Path-connectedness
3.5 Compactness
3.6 Further properties of compactness
4 Identification spaces and cell complexes
4.1 Introduction.
4.2 Final topologies, identification topologies
4.3 Subspaces, products, and identification maps
4.4 Cells and spheres
4.5 Adjunction spaces.
4.6 Properties of adjunction spaces
4.7 Cell complexes
5 Projective and other spaces
5.1 Quaternions
5.2 Normed vector spaces again
5.3 Projective spaces
5.4 Isometries of inner product spaces
5.5 Simplicial complexes
5.6 Bases and sub-bases for open sets; initial topologies
5.7 Joins
5.8 The smash product
5.9 Spaces of functions, and the compact-open topology
6 The fundamental groupoid
6.1 Categories
6.2 Construction of the fundamental groupoid
6.3 Properties of groupoids.
6.4 Functors and morphisms of groupoids
6.5 Homotopies
6.6 Coproducts and pushouts
6.7 The fundamental groupoid of a union of spaces
7 Cofibrations
7.1 The track groupoid
7.2 Fibrations of groupoids.
7.3 Examples
7.4 The gluing theorem for homotopy equivalences of closed unions
7.5 The homotopy type of adjunction spaces
7.6 The cellular approximation theorem
8 Some combinatorial groupoid theory
8.1 Universal morphisms
8.2 Free groupoids
8.3 Quotient groupoids
8.4 Some computations.
9 Computation of the fundamental groupoid
9.1 The Van Kampen theorem for adjunction spaces
9.2 The Jordan Curve Theorem
10 Covering spaces, covering groupoids
10.1 Covering maps and covering homotopies
10.2 Covering groupoids.
10.3 On lifting sums and morphisms
10.4 Existence of covering groupoids
10.5 Lifted topologies
10.6 The equivalence of categories
10.7 Induced coverings and pullbacks
10.8 Applications to subgroup theorems in group theory
11 Orbit spaces, orbit groupoids
11.1 Groups acting on spaces
11.2 Groups acting on groupoids
11.3 General normal subgroupoids and quotient groupoids
11.4 The semidirect product groupoid
11.5 Semidirect product and orbit groupoids
11.6 Full subgroupoids of orbit groupoids
12 Conclusion
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