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Elements of Algebraic Topology by James Munkres
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**About this book :- **
**Elements of Algebraic Topology ** written by
** James Munkres **

This book is intended as a text for a first-year graduate course in algebraic topology; it presents the basic material of homology and cohomology theory.

For students who will go on in topology, differential geometry, Lie groups, or homological algebra, the subject is a prerequisite for later work. For other students, it should be part of their general background, along with algebra and real and complex analysis.

Geometric motivation and applications are stressed throughout. The abstract aspects of the subject are introduced gradually, after the groundwork has been laid with specific examples.

The book begins with a treatment of the simplicial homology groups, the most concrete of the homology theories. After a proof of their topological invariance and verification of the Eilenberg-Steenrod axioms, the singular homology groups are introduced as their natural generalization. CW complexes appear as a useful computational tool. This basic "core" material is rounded out with a treatment of cohomology groups and the cohomology ring.

**Book Detail :- **
** Title: ** Introduction to Topology: Pure and Applied
** Edition: **
** Author(s): ** James Raymond Munkres
** Publisher: ** Perseus Books
** Series: **
** Year: ** 1984
** Pages: ** 465
** Type: ** PDF
** Language: ** English
** ISBN: ** 978-0-2016-2728-2,978-0-2010-4586-4,9780429962462,0429962460
** Country: ** US

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**About Author :- **

Author ** James Raymond Munkres ** (born 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology. Munkres completed his undergraduate education at Nebraska Wesleyan University and received his Ph.D. from the University of Michigan in 1956. He was also elected to the 2018 class of fellows of the American Mathematical Society.

Munkres taught at the University of Michigan and at Princeton University before coming to MIT. His area of research work was Topology, Analysis on Manifolds, Elements of Algebraic Topology, and Elementary Differential Topology. He is also the author of Elementary Linear Algebra.

**All Famous Books of this Author :- **

Here is list all books, text books, editions, versions or solution manuals notes avaliable of this author, We recomended you to download all.

** • Elements of Algebraic Topology by James Munkres **

** • Topología 2.a Edición by James Munkres **

** • Topology 2E by James Munkres **

** • Topology (2E) by James R. Munkres **

** • Topology A First Course by James Munkres **

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**Book Contents :- **
**Elements of Algebraic Topology ** written by
** James Munkres **
cover the following topics.
**1. Homology Groups of a Simplicial Complex **

1 Simplices

2 Simplicial Complexes and Simplicial Maps

3 Abstract Simplicial Complexes

4 Review of Abelian Groups

5 Homology Groups

6 Homology Groups of Surfaces

7 Zero-dimensional Homology

8 The Homology of a Cone

9 Relative Homology

*10 Homology with Arbitrary Coefficients

*11 The Computability of Homology Groups

12 Homomorphisms Induced by Simplicial Maps

13 Chain Complexes and Acyclic Carriers
**2. Topological Invariance of the Homology Groups**

14 Simplicial Approximations

15 Barycentric Subdivision

16 The Simplicial Approximation Theorem

17 The Algebra of Subdivision

18 Topological Invariance of the Homology Groups

19 Homomorphisms Induced by Homotopic Maps

20 Review of Quotient Spaces

*21 Application: Maps of Spheres

*22 Application: The Lefschetz Fixed-point Theorem
**3. Relative Homology and the Eilenberg-Steenrod Axioms**

23 The Exact Homology Sequence

24 The Zig-zag Lemma

25 Mayer-Vietoris Sequences

26 The Eilenberg-Steenrod Axioms

27 The Axioms for Simplicial Theory

*28 Categories and Functors
**4. Singular Homology Theory **

29 The Singular Homology Groups

30 The Axioms for Singular Theory

31 Excision in Singular Homology

*32 Acyclic Models

33 Mayer-Vietoris Sequences

34 The Isomorphism Between Simplicial and Singular Homology

*535 Application: Local Homology Groups and Manifolds

*36 Application: The Jordan Curve Theorem

37 More on Quotient Spaces

38 CW Complexes

39 The Homology of CW Complexes

*40 Application: Projective Spaces and Lens Spaces
**5. Cohomology **

41 The Horn Functor

42 Simplicial Cohomology Groups

43 Relative Cohomology

44 Cohomology Theory

45 The Cohomology of Free Chain Complexes

*46 Chain Equivalences in Free Chain Complexes

47 The Cohomology of CW Complexes

48 Cup Products 285

49 Cohomology Rings of Surfaces
**6. Homology with Coefficients**

50 Tensor Products

51 Homology with Arbitrary Coefficients
**7. Homological Algebra **

52 The Ext Functor

53 The Universal Coefficient Theorem for Cohomology

54 Torsion Products

55 The Universal Coefficient Theorem for Homology

*56 Other Universal Coefficient Theorems

57 Tensor Products of Chain Complexes

58 The Kiinneth Theorem

59 The Eilenberg-Zilber Theorem

*60 The Kiinneth Theorem for Cohomology

*61 Application: The Cohomology Ring of a Product Space
**8. Duality in Manifolds **

62 The Join of Two Complexes

63 Homology Manifolds

64 The Dual Block Complex

65 Poincare Duality

66 Cap Products

67 A Second Proof of Poincare Duality

*68 Application: Cohomology Rings of Manifolds

*69 Application: Homotopy Classification of Lens Spaces

70 Lefschetz Duality

71 Alexander Duality

72 "Natural" Versions of Lefschetz and Alexander Duality

73 Cech Cohomology

74 Alexander-Pontryagin Duality

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