PDF Topologie algébrique Chapitres 1 à 4 Elements De Mathematique French Edition N Bourbaki 9783662493601 Books

Download As PDF : Topologie algébrique Chapitres 1 à 4 Elements De Mathematique French Edition N Bourbaki 9783662493601 Books

Ce livre des Éléments de mathématique est consacré à la Topologie algébrique. Les quatre premiers chapitres présentent la théorie des revêtements d'un espace topologique et du groupe de Poincaré. On construit le revêtement universel d'un espace connexe pointé délaçable et on établit l'équivalence de catégories entre revêtements de cet espace et actions du groupe de Poincaré.

On démontre une version générale du théorème de van Kampen exprimant le groupoïde de Poincaré d'un espace topologique comme un coégalisateur de diagrammes de groupoïdes. Dans de nombreuses situations géométriques, on en déduit une présentation explicite du groupe de Poincaré.

PDF Topologie algébrique Chapitres 1 à 4 Elements De Mathematique French Edition N Bourbaki 9783662493601 Books

"This (very) long review begins with some historical background to set this rather unusual book into context.

The much-delayed publication of this first part of Bourbaki's book on algebraic topology probably caught most by surprise - even if the release in 2012 of the completely rewritten Algèbre: Chapitre 8 (Elements De Mathematique) (French Edition) showed that the pseudonymous author may intend to resume his activities after some 14 years of silence.

As documented both in interviews of earlier generations of Bourbaki members and the publicly available Bourbaki archives, a book on the "stable" (in non-technical sense of the word!) part of algebraic topology was in the plans since the early days of the Éléments de mathématique project. Several outlines and partial drafts were produced starting from 1940's, but despite Bourbaki members such as Samuel Eilenberg and Jean-Pierre Serre, no book materialised for more than 65 years. Later in 1982 when Lie Groups and Lie Algebras: Chapters 7-9 (Elements of Mathematics) was published, the plan had moved to cover the results on covering spaces and homotopy groups needed for Lie theory into a new chapter 11 of General Topology. That did not happen either, though.

One key source of the evident issues the Bourbaki group has had with the book on algebraic topology is the difficulty they have had with incorporating category theory into the treatise. As is amply documented, just as category theory was starting to develop (in the context of algebraic topology!), Bourbaki decided on using purely set-theoretic "structures" as the common basis for his series of books. Bourbaki's structures are a more limited framework and, in particular in the fields of algebraic topology and homological algebra, developing the theory without category theory is a clear handicap.

A book on category theory (and homological algebra in Abelian categories) was planned and referenced in 1961 in the first chapter of Commutative Algebra: Chapters 1-7. However, around that same time a decision was made that despite the advantages of thorough incorporation of category theory into the series, it would require too significant rewriting of existing volumes (André Weil is given in some sources as the key person behind the decision - despite having officially retired from the group at the time). The result was that the later Bourbaki books exhibit an uneasy accommodation of categorical thinking and sometimes even terminology (e.g., functors appear is headings) while technically no category theory is used. Predictably the treatment if homological algebra (Algèbre: Chapitre 10. Algèbre homologique (Elements De Mathematique) (French Edition) from 1982) suffers from the reduced scope due to the unwillingness to go properly functorial.

It is into this context that this new book about one of the most category-heavy major branches of mathematics is born. The scope is much expanded from merely one new chapter of General Topology. And categories make their appearance! - however, in a limited manner, mainly to introduce groupoids and not (at least in the first four chapters) to provide an overall framework for the theory being developed. Instead, categorical approach very much underlies the presentation while it is still almost entirely written without categorical language (there is one paragraph in chapter 4 where for the benefit of a reader knowledgeable of category theory it is stated that the fibre functor is an equivalence of categories from covering spaces to sets with action of the fundamental group).

Chapter I of the book is a detailed (150 pages) treatment of covering spaces from "general topology" viewpoint, before introduction of homotopy and fundamental groups. The subject matter is developed in a way that parallels the development of similar concepts of algebraic geometry in Grothendieck's EGA and SGA. The chapter starts with a thorough discussion of fibre products, strict and universally strict morphisms in the category of topological spaces, and continues with the introduction of "separated morphisms" (a relativised version of Hausdorff separation axiom copied from the theory of schemes) and local homeomorphisms.

Next section introduces sheaves - another long-planned topic for the Bourbaki series! Topics covered include the étalé space of a sheaf (which is used to construct the sheaf associated to a presheaf), inverse and direct images of sheaves and their adjoint property, soft sheaves and sheaves with algebraic structure (groups, rings). However, neither kernels nor cokernels of morphisms and hence exact sequences of sheaves nor sheaves of modules over a sheaf of rings are covered. Next, covering spaces and their relation to locally constant sheaves is covered in detail, followed by principal bundles (aka torsors). Finally, universal covering spaces are defined and simply connected spaces are defined as the ones having only trivialisable covering spaces.

Chapter 2 is a rather idiosyncratic treatment of groupoids. As discussed above, categories are defined but essentially only for the purposes of developing enough theory of groupoids for the purposes of the fundamental groupoid of a space and a groupoid-based proof of the Seifert-van Kampen theorem. Bourbaki first introduces quivers and graphs (no real theorems here). Categories are defined as quivers with composition of arrows. Functors are also defined but natural transformations are not (!). That "structures" in the sense of Bourbaki define categories is given as an example. Then follows a detailed discussion of groupoids from a fairly combinatorial viewpoint.

Next homotopies of functors between groupoids are defined as natural transformations. What is done here likely appears as ad hoc to many readers since the context is completely missing - this is just the restriction to groupoids of homotopies in the classical model structure on the category of (small) categories. It gets more curious with the introduction of homotopy coequalisers (homotopy colimits of a pair of morphisms) for groupoids, again with no theoretical context to help explain what is being done. After this (ordinary) coequalisers are discussed and the canonical morphism from homotopy colimit to ordinary colimit is discussed, specialised to coequalisers of groupoids. These developments are used later in chapter 4 to translate Seifert-van Kampen from groupoids to fundamental groups.

Chapter 3 introduces homotopy for topological spaces. The treatment is conventional, with homotopies treated as "left homotopy" (via cylinder constructions). Mapping cylinders, mapping cones and cofibrations are defined (the latter as "pairs having the homotopy extension property"). However, the "right homotopy" viewpoint is essentially missing: no mapping path spaces, homotopy fibres or discussions of (Hurewicz) fibrations. As a related point, Bourbaki does not use compactly generated spaces or other such category that would be Cartesian closed.

Next paths, path connected and locally path-connected spaces are discussed, followed by a few criteria for lifting of paths. Then follows a section on fundamental group and groupoid of spaces. Finally, homotopy and path lifting properties of covering spaces are discussed, finishing with the case of locally path-connected spaces and actions of fundamental group on the fibre of a covering.

Chapter 4 develops the theory for spaces that have universal covering space (locally connected spaces that are semi-locally simply connected - Bourbaki has coined the term "délaçable" - an amusing reference to undoing shoe laces or delacing a corset...).
After construction of the universal coveringspace, implications on fundamental group and classification of covering spaces - together with the already mentioned statement of equivalence of categories between covering spaces and (discrete) G-sets where G is the fundamental group of the base. There is also a separate section on fundamental groups of topological groups.

Next comes another surprise topic: Grothendieck's descent theory from algebraic geometry adapted for topological spaces. This theory (essentially a generalisation of "glueing" together of objects defined locally) is developed with a view towards descent of groupoids, to be applied next to fundamental groupoids to prove the groupoid version of Seifert-van Kampen theorem. The chapter ends with an introduction to classifying spaces of topological groups and the construction of classifying spaces for discrete groups.

Like all Bourbaki books (apart from the "fascicules de résultats"), the one under review contains extensive exercises (72 pages).

At 513 pages the book is long and detailed. With some (unfair!) exaggeration, the list of topics covered is comparable to the first 46 pages of Peter May's wonderful (and dense) A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics Series). The introduction to the volume under review states that forthcoming chapters will deal with homology, cohomology, higher homotopy groups and CW complexes. In addition, a list of updated references for the book on Lie Groups and Algebras shows that elementary results on higher homotopy groups of spheres will appear in chapter 7. Assuming that comparable level of detail will continue to apply to the forthcoming chapters, there will be many more pages to come!

This reviewer thinks that Bourbaki's new book on algebraic topology is a welcome addition to the literature. Like all Bourbaki volumes, it gives very careful and complete proofs of everything, and the clarity of exposition is excellent. The decision to treat the Seifert-van Kampen theorem using groupoids is a good one, as is the introduction to descent theory in the context of topological spaces. Chapter 1 indeed feels like it could have been an excellent fit into the author's book on general topology. This volume will serve as a good reference, and parts of it could be used by students to learn the material (and in particular to supplement other books that leave many "obvious" proofs to be provided by the reader.

On the negative side, it is undeniable that the book suffers from the implicit-only usage made of category theory. First, it prevents the development of many topics in their natural context and generality and leads to some definitions that will appear to be ad hoc to many readers - homotopy of groupoids and homotopy limits have been discussed above. In the chapters to come this limitation will likely show up even more clearly. Second, it will likely limit the topics that can be covered - conceptual clarity from Quillen's model categories, total derived functors, homotopy limits and colimits and similar topics will likely be out of scope, which if true will be a missed opportunity for Bourbaki. Also treatment of simplicial homotopy theory could be out of scope for the same reasons. It will be interesting to see how Bourbaki will deal with (co)homology given the categorical limitations.

These issues of course originate in the long-past decision not to "categorise" the Éléments de mathématique series. Given this constraint, it is actually quite surprising how smoothly the author manages to develop the material for a reader with background in category theory."

Product details Series Elements De Mathematique Paperback 498 pages Publisher Springer; 1ère éd. 2016 edition (April 30, 2016) Language French ISBN-10 9783662493601 ISBN-13 978-3662493601 ASIN 3662493608

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Topologie algébrique Chapitres 1 à 4 Elements De Mathematique French Edition N Bourbaki 9783662493601 Books Reviews :

Topologie algébrique Chapitres 1 à 4 Elements De Mathematique French Edition N Bourbaki 9783662493601 Books Reviews

• This (very) long review begins with some historical background to set this rather unusual book into context.
  
  The much-delayed publication of this first part of Bourbaki's book on algebraic topology probably caught most by surprise - even if the release in 2012 of the completely rewritten Algèbre Chapitre 8 (Elements De Mathematique) (French Edition) showed that the pseudonymous author may intend to resume his activities after some 14 years of silence.
  
  As documented both in interviews of earlier generations of Bourbaki members and the publicly available Bourbaki archives, a book on the "stable" (in non-technical sense of the word!) part of algebraic topology was in the plans since the early days of the Éléments de mathématique project. Several outlines and partial drafts were produced starting from 1940's, but despite Bourbaki members such as Samuel Eilenberg and Jean-Pierre Serre, no book materialised for more than 65 years. Later in 1982 when Lie Groups and Lie Algebras Chapters 7-9 (Elements of Mathematics) was published, the plan had moved to cover the results on covering spaces and homotopy groups needed for Lie theory into a new chapter 11 of General Topology. That did not happen either, though.
  
  One key source of the evident issues the Bourbaki group has had with the book on algebraic topology is the difficulty they have had with incorporating category theory into the treatise. As is amply documented, just as category theory was starting to develop (in the context of algebraic topology!), Bourbaki decided on using purely set-theoretic "structures" as the common basis for his series of books. Bourbaki's structures are a more limited framework and, in particular in the fields of algebraic topology and homological algebra, developing the theory without category theory is a clear handicap.
  
  A book on category theory (and homological algebra in Abelian categories) was planned and referenced in 1961 in the first chapter of Commutative Algebra Chapters 1-7. However, around that same time a decision was made that despite the advantages of thorough incorporation of category theory into the series, it would require too significant rewriting of existing volumes (André Weil is given in some sources as the key person behind the decision - despite having officially retired from the group at the time). The result was that the later Bourbaki books exhibit an uneasy accommodation of categorical thinking and sometimes even terminology (e.g., functors appear is headings) while technically no category theory is used. Predictably the treatment if homological algebra (Algèbre Chapitre 10. Algèbre homologique (Elements De Mathematique) (French Edition) from 1982) suffers from the reduced scope due to the unwillingness to go properly functorial.
  
  It is into this context that this new book about one of the most category-heavy major branches of mathematics is born. The scope is much expanded from merely one new chapter of General Topology. And categories make their appearance! - however, in a limited manner, mainly to introduce groupoids and not (at least in the first four chapters) to provide an overall framework for the theory being developed. Instead, categorical approach very much underlies the presentation while it is still almost entirely written without categorical language (there is one paragraph in chapter 4 where for the benefit of a reader knowledgeable of category theory it is stated that the fibre functor is an equivalence of categories from covering spaces to sets with action of the fundamental group).
  
  Chapter I of the book is a detailed (150 pages) treatment of covering spaces from "general topology" viewpoint, before introduction of homotopy and fundamental groups. The subject matter is developed in a way that parallels the development of similar concepts of algebraic geometry in Grothendieck's EGA and SGA. The chapter starts with a thorough discussion of fibre products, strict and universally strict morphisms in the category of topological spaces, and continues with the introduction of "separated morphisms" (a relativised version of Hausdorff separation axiom copied from the theory of schemes) and local homeomorphisms.
  
  Next section introduces sheaves - another long-planned topic for the Bourbaki series! Topics covered include the étalé space of a sheaf (which is used to construct the sheaf associated to a presheaf), inverse and direct images of sheaves and their adjoint property, soft sheaves and sheaves with algebraic structure (groups, rings). However, neither kernels nor cokernels of morphisms and hence exact sequences of sheaves nor sheaves of modules over a sheaf of rings are covered. Next, covering spaces and their relation to locally constant sheaves is covered in detail, followed by principal bundles (aka torsors). Finally, universal covering spaces are defined and simply connected spaces are defined as the ones having only trivialisable covering spaces.
  
  Chapter 2 is a rather idiosyncratic treatment of groupoids. As discussed above, categories are defined but essentially only for the purposes of developing enough theory of groupoids for the purposes of the fundamental groupoid of a space and a groupoid-based proof of the Seifert-van Kampen theorem. Bourbaki first introduces quivers and graphs (no real theorems here). Categories are defined as quivers with composition of arrows. Functors are also defined but natural transformations are not (!). That "structures" in the sense of Bourbaki define categories is given as an example. Then follows a detailed discussion of groupoids from a fairly combinatorial viewpoint.
  
  Next homotopies of functors between groupoids are defined as natural transformations. What is done here likely appears as ad hoc to many readers since the context is completely missing - this is just the restriction to groupoids of homotopies in the classical model structure on the category of (small) categories. It gets more curious with the introduction of homotopy coequalisers (homotopy colimits of a pair of morphisms) for groupoids, again with no theoretical context to help explain what is being done. After this (ordinary) coequalisers are discussed and the canonical morphism from homotopy colimit to ordinary colimit is discussed, specialised to coequalisers of groupoids. These developments are used later in chapter 4 to translate Seifert-van Kampen from groupoids to fundamental groups.
  
  Chapter 3 introduces homotopy for topological spaces. The treatment is conventional, with homotopies treated as "left homotopy" (via cylinder constructions). Mapping cylinders, mapping cones and cofibrations are defined (the latter as "pairs having the homotopy extension property"). However, the "right homotopy" viewpoint is essentially missing no mapping path spaces, homotopy fibres or discussions of (Hurewicz) fibrations. As a related point, Bourbaki does not use compactly generated spaces or other such category that would be Cartesian closed.
  
  Next paths, path connected and locally path-connected spaces are discussed, followed by a few criteria for lifting of paths. Then follows a section on fundamental group and groupoid of spaces. Finally, homotopy and path lifting properties of covering spaces are discussed, finishing with the case of locally path-connected spaces and actions of fundamental group on the fibre of a covering.
  
  Chapter 4 develops the theory for spaces that have universal covering space (locally connected spaces that are semi-locally simply connected - Bourbaki has coined the term "délaçable" - an amusing reference to undoing shoe laces or delacing a corset...).
  After construction of the universal coveringspace, implications on fundamental group and classification of covering spaces - together with the already mentioned statement of equivalence of categories between covering spaces and (discrete) G-sets where G is the fundamental group of the base. There is also a separate section on fundamental groups of topological groups.
  
  Next comes another surprise topic Grothendieck's descent theory from algebraic geometry adapted for topological spaces. This theory (essentially a generalisation of "glueing" together of objects defined locally) is developed with a view towards descent of groupoids, to be applied next to fundamental groupoids to prove the groupoid version of Seifert-van Kampen theorem. The chapter ends with an introduction to classifying spaces of topological groups and the construction of classifying spaces for discrete groups.
  
  Like all Bourbaki books (apart from the "fascicules de résultats"), the one under review contains extensive exercises (72 pages).
  
  At 513 pages the book is long and detailed. With some (unfair!) exaggeration, the list of topics covered is comparable to the first 46 pages of Peter May's wonderful (and dense) A Concise Course in Algebraic Topology (Chicago Lectures in Mathematics Series). The introduction to the volume under review states that forthcoming chapters will deal with homology, cohomology, higher homotopy groups and CW complexes. In addition, a list of updated references for the book on Lie Groups and Algebras shows that elementary results on higher homotopy groups of spheres will appear in chapter 7. Assuming that comparable level of detail will continue to apply to the forthcoming chapters, there will be many more pages to come!
  
  This reviewer thinks that Bourbaki's new book on algebraic topology is a welcome addition to the literature. Like all Bourbaki volumes, it gives very careful and complete proofs of everything, and the clarity of exposition is excellent. The decision to treat the Seifert-van Kampen theorem using groupoids is a good one, as is the introduction to descent theory in the context of topological spaces. Chapter 1 indeed feels like it could have been an excellent fit into the author's book on general topology. This volume will serve as a good reference, and parts of it could be used by students to learn the material (and in particular to supplement other books that leave many "obvious" proofs to be provided by the reader.
  
  On the negative side, it is undeniable that the book suffers from the implicit-only usage made of category theory. First, it prevents the development of many topics in their natural context and generality and leads to some definitions that will appear to be ad hoc to many readers - homotopy of groupoids and homotopy limits have been discussed above. In the chapters to come this limitation will likely show up even more clearly. Second, it will likely limit the topics that can be covered - conceptual clarity from Quillen's model categories, total derived functors, homotopy limits and colimits and similar topics will likely be out of scope, which if true will be a missed opportunity for Bourbaki. Also treatment of simplicial homotopy theory could be out of scope for the same reasons. It will be interesting to see how Bourbaki will deal with (co)homology given the categorical limitations.
  
  These issues of course originate in the long-past decision not to "categorise" the Éléments de mathématique series. Given this constraint, it is actually quite surprising how smoothly the author manages to develop the material for a reader with background in category theory.