Algebraic topology prerequisites

EMS Textbooks in Mathematics

Tammo tom Dieck (University of Göttingen, Germany)

Algebraic Topology

Corrected 2nd printing, 2010

ISBN print 978-3-03719-048-7, ISBN online 978-3-03719-548-2
DOI 10.4171/048
September 2008, 578 pages, hardcover, 16.5 x 23.5 cm.
58.00 Euro

This book is written as a textbook on algebraic topology. The first part covers the material for two introductory courses about homotopy and homology. The second part presents more advanced applications and concepts (duality, characteristic classes, homotopy groups of spheres, bordism). The author recommends to start an introductory course with homotopy theory. For this purpose, classical results are presented with new elementary proofs. Alternatively, one could start more traditionally with singular and axiomatic homology. Additional chapters are devoted to the geometry of manifolds, cell complexes and fibre bundles. A special feature is the rich supply of nearly 500 exercises and problems. Several sections include topics which have not appeared before in textbooks as well as simplified proofs for some important results.

Prerequisites are standard point set topology (as recalled in the first chapter), elementary algebraic notions (modules, tensor product), and some terminology from category theory. The aim of the book is to introduce advanced undergraduate and graduate (masters) students to basic tools, concepts and results of algebraic topology. Sufficient background material from geometry and algebra is included.

Keywords: Covering spaces, fibrations, cofibrations, homotopy groups, cell complexes, fibre bundles, vector bundles, classifying spaces, singular and axiomatic homology and cohomology, smooth manifolds, duality, characteristic classes, bordism.

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Further Information

Review in Zentralblatt MATH 1156.55001

Review in MR2456045 (2009f:55001)

EMS Review

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