Algebraic Topology by Allen Hatcher

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

About this book :-
"Algebraic Topology" by "Allen Hatcher" is a classic and highly respected textbook in modern mathematics. It introduces algebraic topology by showing how algebraic methods can be used to understand the shape and structure of topological spaces. The book is known for its clear explanations, geometric intuition, and accessible writing style. The text covers essential topics such as "homotopy", "homology", and "cohomology", building concepts step by step with many examples and illustrations. Hatcher focuses on helping readers develop intuition rather than memorizing formulas. This approach makes the book especially valuable for students learning advanced "topology" for the first time. Overall, the book is widely used by graduate students and advanced undergraduates around the world. Its clear structure and depth make it a long-term reference in "pure mathematics". Freely available online, it has become a standard resource for anyone studying "algebraic topology" in a serious academic setting.

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: University Educational Resource
Amazon: Amazon

About Author :-
The author Allen Hatcher was born in Indiana and received his Ph.D. in 1971 from Stanford University. He is a highly respected mathematician known for his work in topology and his impact on mathematics education. He spent much of his academic career at Cornell University, where he taught and researched advanced mathematical concepts. Hatcher is especially known for explaining complex ideas with clarity and intuition. His work has shaped modern "algebraic topology" and influenced teaching in "pure mathematics". Through his writing, he has made topics like "homotopy" and "homology" more accessible to students worldwide.

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