Intended for use both as a text and a reference, this book is an exposition of the fundamental ideas of algebraic topology. The first third of the book covers the fundamental group, its definition and its application in the study of covering spaces. The focus then turns to homology theory, including cohomology, cup products, cohomology operations, and topological manifolds. The remaining third of the book is devoted to Homotropy theory, covering basic facts about homotropy groups, applications to obstruction theory, and computations of homotropy groups of spheres. In the later parts, the main emphasis is on the application to geometry of the algebraic tools developed earlier.
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This is the classical introduction to algebraic topology. However, because of the tight writing style, the book serves better as a reference than as a text. Read full review
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A C X abelian groups acyclic algebra base point base space chain complex chain equivalence chain map closed path cochain coefficients cofibration compact composite continuous map corresponding covariant functor covering projection covering space CW complex defined deformation retract denoted edge path element epimorphism example fibration fibration with unique finite follows from theorem fundamental group Given Hausdorff space homology group homology theory homomorphism homotopy category homotopy classes homotopy equivalence homotopy type Horn inclusion map inverse lemma Let Let F Let G locally path connected map F module morphisms n-simplex neighborhood nonempty open covering open subset oriented path component path-connected space pointed space polyhedron presheaf proof Let prove relative CW complex short exact sequence simplex simplicial approximation simplicial complex simplicial map simply connected singular homology spectral sequence strong deformation retract subcomplex subdivision subspace theorem Let topological space unique path lifting vertex vertices