Algebraic Topology by Allen Hatcher

About this book :-
Allen Hatcher's Algebraic Topology is a widely used textbook that introduces the field of algebraic topology. In most major universities it's often used in graduate-level courses and is appreciated for its clear, geometric approach. This introductory textbook in algebraic topology is suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. The book covers topics like fundamental groups, covering spaces, and homology, providing both theoretical explanations and practical examples. However, some readers find the style informal and occasionally lacking in rigorous detail, which can make certain concepts challenging to grasp. Despite this, many students and instructors value it as a comprehensive resource for learning algebraic topology. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers.

Book Detail :-
Title: Algebraic Topology by Allen Hatcher
Publisher: Cambridge University Press
Year: 2001
Pages: 556
Type: PDF
Language: English
ISBN-10 #: 0521795400
ISBN-13 #: 978-0521795401
License: Linked Content Owned by Author
Amazon: Amazon

About Author :-
The author Allen Hatcher was born in Indiana. After obtain his B.S from Oberlin College in 1966, he went for his graduate studies to Stanford University, where he received his Ph.D. in 1971. Hatcher start his career in Princeton University Assistant Professor from 1973 to 1979. He was also a member of the Institute for Advanced Study in 1975–76 and 1979–80. Hatcher went on to become a professor at the University of California, Los Angeles in 1977. From 1983 he has been a professor at Cornell University; he is now a professor emeritus. In 1978 Hatcher was an invited speaker at the International Congresses of Mathematicians in Helsinki.

Book Contents :-
0. Some Underlying Geometric Notions, Homotopy and Homotopy Type. Cell Complexes. Operations on Spaces. Two Criteria for Homotopy Equivalence. The Homotopy Extension Property. 1. Fundamental Group and Covering Spaces, Basic Constructions, Paths and Homotopy. The Fundamental Group of the Circle. Induced Homomorphisms, Van Kampen's Theorem, Free Products of Groups. The van Kampen Theorem. Applications to Cell Complexes, Covering Spaces, Lifting Properties. The Classification of Covering Spaces. Deck Transformations and Group Actions, Additional Topics, Graphs and Free Groups. K(G,1) Spaces and Graphs of Groups. 2. Homology, Simplicial and Singular Homology, Delta-Complexes. Simplicial Homology. Singular Homology. Homotopy Invariance. Exact Sequences and Excision. The Equivalence of Simplicial and Singular Homology, Computations and Applications, Degree. Cellular Homology. Mayer-Vietoris Sequences. Homology with Coefficients, The Formal Viewpoint, Axioms for Homology. Categories and Functors, Additional Topics, Homology and Fundamental Group. Classical Applications. Simplicial Approximation. 3. Cohomology, Cohomology Groups, The Universal Coefficient Theorem. Cohomology of Spaces, Cup Product, The Cohomology Ring. A Kunneth Formula. Spaces with Polynomial Cohomology, Poincare Duality, orientations and Homology. The Duality Theorem. Cup Product and Duality. Other Forms of Duality, Additional Topics, The Universal Coefficient Theorem for Homology. The General Kunneth Formula. H-Spaces and Hopf Algebras. The Cohomology of SO(n). Bockstein Homomorphisms. Limits. More About Ext. Transfer Homomorphisms. Local Coefficients 4. Homotopy Theory, Homotopy Groups, Definitions and Basic Constructions. Whitehead's Theorem. Cellular Approximation. CW Approximation, Elementary Methods of Calculation, Excision for Homotopy Groups. The Hurewicz Theorem. Fiber Bundles. Stable Homotopy Groups, Connections with Cohomology, The Homotopy Construction of Cohomology. Fibrations. Postnikov Towers. Obstruction Theory, Additional Topics, Basepoints and Homotopy. The Hopf Invariant. Minimal Cell Structures. Cohomology of Fiber Bundles. The Brown Representability Theorem. Spectra and Homology Theories. Gluing Constructions. Eckmann-Hilton Duality. Stable Splittings of Spaces. The Loopspace of a Suspension. Symmetric Products and the Dold-Thom Theorem. Steenrod Squares and Powers

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