By Tom Dieck T.

This ebook is written as a textbook on algebraic topology. the 1st half covers the cloth for 2 introductory classes approximately homotopy and homology. the second one half offers extra complicated functions and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy thought. For this goal, classical effects are awarded with new straight forward proofs. then again, you may begin extra routinely with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, mobilephone complexes and fibre bundles. a distinct function is the wealthy provide of approximately 500 routines and difficulties. a number of sections comprise issues that have now not seemed sooner than in textbooks in addition to simplified proofs for a few vital effects. necessities are regular aspect set topology (as recalled within the first chapter), simple algebraic notions (modules, tensor product), and a few terminology from classification thought. the purpose of the ebook is to introduce complex undergraduate and graduate (master's) scholars to easy instruments, thoughts and result of algebraic topology. adequate history fabric from geometry and algebra is incorporated. A ebook of the ecu Mathematical Society (EMS). allotted in the Americas by means of the yankee Mathematical Society.

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**Extra resources for Algebraic topology**

**Sample text**

What is the sum of two objects in TOPK ? What is the product of two objects in TOPB ? 13. A pullback of a shrinkable map is shrinkable. A pushout of a deformation retract is a deformation retract. 14. Let Y; Z be compact or X; Z be locally compact. X ^ Y / ^ Z ! Y ^ Z/ is a homeomorphism. ) the map is always a homeomorphism. ) 15. Aj ^ B/ ! canonical map which is on each sumj Aj ^ B be theW mand Ak ^ B induced by the inclusion Ak ! j Aj . Show that this map is a homeomorphism if the index set is finite.

10. Construct an inclusion A X which is a retract and a homotopy equivalence but not a deformation retract. 11. Construct a map p W E ! b/ are contractible but which does not have a section. Construct an h-equivalence p W E ! B which has a section but which is not shrinkable. 12. What is the sum of two objects in TOPK ? What is the product of two objects in TOPB ? 13. A pullback of a shrinkable map is shrinkable. A pushout of a deformation retract is a deformation retract. 14. Let Y; Z be compact or X; Z be locally compact.

Y has a right homotopy inverse h W Y ! , f h ' id, and a left homotopy inverse g W Y ! , gf ' id, then f is an h-equivalence. If two of the maps f W X ! Y , g W Y ! Z, and gf are h-equivalences, then so isQ the third. Homotopy is compatible with sums and products. Let pi W j 2J Xj ! Xi be the projection onto the i -th factor. Then Q Q ŒY; j 2J Xj ! j 2J ŒY; Xj ; Œf 7! Œpi ı f / ` is a well-defined bijection. Let ik W Xk ! j 2J Xj be the canonical inclusion of the k-th summand. Then Q ` Œ j 2J Xj ; Y !