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Simplicial Objects in Algebraic Topology

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Simplicial Objects in Algebraic Topology Cover

 

Synopses & Reviews

Publisher Comments:

Simplicial sets are discrete analogs of topological spaces. They have played a central role in algebraic topology ever since their introduction in the late 1940s, and they also play an important role in other areas such as geometric topology and algebraic geometry. On a formal level, the homotopy theory of simplicial sets is equivalent to the homotopy theory of topological spaces. In view of this equivalence, one can apply discrete, algebraic techniques to perform basic topological constructions. These techniques are particularly appropriate in the theory of localization and completion of topological spaces, which was developed in the early 1970s.

Since it was first published in 1967, Simplicial Objects in Algebraic Topology has been the standard reference for the theory of simplicial sets and their relationship to the homotopy theory of topological spaces. J. Peter May gives a lucid account of the basic homotopy theory of simplicial sets, together with the equivalence of homotopy theories alluded to above. The central theme is the simplicial approach to the theory of fibrations and bundles, and especially the algebraization of fibration and bundle theory in terms of "twisted Cartesian products." The Serre spectral sequence is described in terms of this algebraization. Other topics treated in detail include Eilenberg-MacLane complexes, Postnikov systems, simplicial groups, classifying complexes, simplicial Abelian groups, and acyclic models.

 

"Simplicial Objects in Algebraic Topology presents much of the elementary material of algebraic topology from the semi-simplicial viewpoint. It should prove very valuable to anyone wishing to learn semi-simplicial topology. [May] has included detailed proofs, and he has succeeded very well in the task of organizing a large body of previously scattered material."—Mathematical Review

About the Author

J. Peter May is professor of mathematics at the University of Chicago.

Table of Contents

I. SIMPLICIAL OBJECTS AND HOMOTOPY   

1. Definitions  and  examples

2. Simplicial objects in categories; homology

3. Homotopy of Kan complexes

4. The group structures

5. Homotopy of simplicial maps

6. Function complexes  

Bibliographical notes on chapter I

II. FIBRATIONS, POSTNIKOV SYSTEMS, AND MINIMAL COMPLEXES    

7. Kan  fibrations 

8. Postnikov systems

9. Minimal complexes

10. Minimal fibrations 

11. Fibre products and fibre bundles

12. Weak homotopy type  

13. The Hurewicz  theorems

Bibliographical notes on chapter II 

III. GEOMETRIC REALIZATION    

14. The realization   

15. Adjoint functors 

16. Comparison of simplicial sets and topological spaces

Bibliographical notes on chapter III

IV. TWISTED CARTESIAN PRODUCTS AND FIBRE BUNDLES

17. Simplicial groups

18. Principal fibrations and twisted Cartesian products

19. The  group  of a  fibre  bundle

20. Fibre bundles and twisted Cartesian products 

21. Universal bundles and classifying complexes 

Bibliographical notes on chapter IV

V. EILENBERG-MACLANE COMPLEXES AND POSTNIKOV  SYSTEMS

22. Simplicial Abelian  groups

23. Eilenberg-MacLane complexes

24. K(rr, n)'s and cohomology operations

25. The k-invariants of Postnikov systems

Bibliographical notes on chapter V

VI. LOOP GROUPS, ACYCLIC MODELS, AND TWISTED TENSOR PRODUCTS

26.  Loop  groups 

27. The  functors G, W, and  E 

28.  Acyclic  models

29. The Eilenberg-Zilber theorem

30. Cup, Pontryagin, and cap products; twisting cochains

31. Brown's  theorem 

32. The Serre  spectral sequence 

Bibliographical notes on chapter VI

BIBLIOGRAPHY

Product Details

ISBN:
9780226511818
Author:
May, J. Peter
Publisher:
University of Chicago Press
Author:
May, J. P.
Subject:
General
Subject:
Mathematics
Subject:
Algebra
Subject:
Algebra - General
Subject:
General Mathematics
Subject:
Mathematics - General
Edition Description:
1
Series:
Chicago Lectures in Mathematics
Publication Date:
19930131
Binding:
TRADE PAPER
Grade Level:
Professional and scholarly
Language:
English
Pages:
170
Dimensions:
8.50x5.58x.44 in. .47 lbs.

Related Subjects


Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Topology

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