Introduction to Homotopy Theory
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|Format: ||Paperback, 344 pages|
|Other Information: ||Illustrated|
|Published In: ||United States, 01 July 2011|
This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic Homotopy; H-spaces and co-H-spaces; fibrations and cofibrations; exact sequences of homotopy sets, actions, and coactions; homotopy pushouts and pullbacks; classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead; homotopy Sets; homotopy and homology decompositions of spaces and maps; and obstruction theory. The underlying theme of the entire book is the Eckmann-Hilton duality theory. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.
Table of Contents
1 Basic Homotopy.- 1.1 Introduction.- 1.2 Spaces, Maps, Products and Wedges.- 1.3 Homotopy I.- 1.4 Homotopy II.- 1.5 CW Complexes.- 1.6 Why Study Homotopy Theory?.- Exercises.- 2 H-Spaces and Co-H-Spaces.- 2.1 Introduction.- 2.2. H-Spaces and Co-H-Spaces.- 2.3 Loop Spaces and Suspensions.- 2.4 Homotopy Groups I.- 2.5 Moore Spaces and Eilenberg-Mac Lane Spaces.- 2.6 Eckmann-Hilton Duality I.- Exercises.- 3 Cofibrations and Fibrations.- 3.1 Introduction.- 3.2 Cofibrations.- 3.3 Fibrations.- 3.4 Examples of Fiber Bundles.- 3.5 Replacing a Map by a Cofiber or Fiber Map.- Exercises.- 4 Exact Sequences.- 4.1 Introduction.- 4.2 The Coexact and Exact Sequence of a Map.- 4.3 Actions and Coactions.- 4.4 Operations.- 4.5 Homotopy Groups II.- Exercises.- 5 Applications of Exactness.- 5.1 Introduction.- 5.2 Universal Coefficient Theorems.- 5.3 Homotopical Cohomology Groups.- 5.4 Applications to Fiber and Cofiber Sequences.- 5.5 The Operation of the Fundamental Group.- 5.6 Calculation of Homotopy Groups.-Exercises.- 6 Homotopy Pushouts and Pullbacks.- 6.1 Introduction.- 6.2 Homotopy Pushouts and Pullbacks I.- 6.3 Homotopy Pushouts and Pullbacks II.- 6.4 Theorems of Serre, Hurewicz and Blakers-Massey.- 6.5 Eckmann-Hilton Duality II.- Exercises.- 7 Homotopy and Homology Decompositions.- 7.1 Introduction.- 7.2 Homotopy Decompositions of Spaces.- 7.3 Homology Decompositions of Spaces.- 7.4 Homotopy and Homology Decompositions of Maps.- Exercises.- 8 Homotopy Sets.- 8.1 Introduction.- 8.2 The Set [X, Y].- 8.3 Category.- 8.4 Loop and Group Structure in [X, Y].-Exercises.- 9 Obstruction Theory.- 9.1 Introduction.- 9.2 Obstructions Using Homotopy Decompositions.- 9.3 Lifts and Extensions.- 9.4 Obstruction Miscellany.- Exercises.- A Point-Set Topology.- B The Fundamental Group.- C Homology and Cohomology.- D Homotopy Groups of the n-Sphere.- E Homotopy Pushouts and Pullbacks.- F Categories and Functors.- Hints to Some of the Exercises.- References.- Index.-
About the Author
Martin Arkowitz is currently a professor of mathematics at Dartmouth College. He received his Ph.D. in mathematics at Cornell University. His area of expertise is algebraic topology.
From the reviews: "Homotopy theory constitutes a branch of algebraic topology, a subject whose modus operandi, enshrined in its very name, consists of attaching algebraic objects to topological spaces for the sake of reducing topological problems to simpler algebraic ones. ! Summing Up: Recommended. Upper-division undergraduates and above." (D. V. Feldman, Choice, Vol. 49 (7), March, 2012)
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