Algebraic Topology

More about the book

This comprehensive work delves into the realms of analytic and algebraic topology, presenting a structured exploration of simplicial complexes, chain complexes, and cell complexes. The introduction sets the stage, followed by an in-depth examination of simplicial complexes, including their geometry, homology, and cohomology groups, with sections dedicated to orientation, incidence numbers, and practical examples. The discussion extends into chain complexes, covering their general theory, subcomplexes, and boundary operators, alongside free chain complexes and their canonical bases. The focus then shifts to cell complexes, detailing cell decompositions and the homology groups associated with them, while emphasizing the invariance of these groups through rigorous proofs and generalizations. The latter sections advance the theory, exploring products in polyhedra and manifolds, including definitions, complementary cell decompositions, and the pivotal Poincaré Duality Theorem. The work culminates in a thorough analysis of the cohomology ring of a manifold, addressing products and product matrices. The bibliography and index provide essential resources for further study, making this text an invaluable reference for those engaged in the study of topology.