This English translation of a Russian book presents the basic notions of differential and algebraic topology, which are indispensable for specialists and useful for research mathematicians and theoretical physicists. In particular, ideas and results are introduced related to manifolds, cell spaces, coverings and fibrations, homotopy groups, intersection index, etc. The author notes, 'The lecture note origins of the book left a significant imprint on its style. It contains very few detailed proofs: I tried to give as many illustrations as possible and to show what really occurs in topology, not always explaining why it occurs'. He concludes, 'As a rule, only those proofs (or sketches of proofs) that are interesting per se and have important generalizations are presented'.
Topological spaces and operations with them Homotopy groups and homotopy equivalence Coverings Cell spaces ($CW$-complexes) Relative homotopy groups and the exact sequence of a pair Fiber bundles Smooth manifolds The degree of a map Homology: Basic definitions and examples main properties of singular homology groups and their computation Homology of cell spaces Morse theory Cohomology and Poincare duality Some applications of homology theory Multiplication in cohomology (and homology) Index of notations Subject index.