This textbook on algebraic topology updates a popular textbook from
the golden era of the Moscow school of I. M. Gelfand. The first
English translation, done many decades ago, remains very much in
demand, although it has been long out-of-print and is difficult to
obtain. Therefore, this updated English edition will be much
welcomed by the mathematical community. Distinctive features of
this book include: a concise but fully rigorous presentation,
supplemented by a plethora of illustrations of a high technical and
artistic caliber; a huge number of nontrivial examples and
computations done in detail; a deeper and broader treatment of
topics in comparison to most beginning books on algebraic topology;
an extensive, and very concrete, treatment of the machinery of
spectral sequences. The second edition contains an entirely new
chapter on K-theory and the Riemann-Roch theorem (after Hirzebruch
and Grothendieck).
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