By Tammo Tom Dieck

This booklet 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 provides extra complex purposes and ideas (duality, attribute periods, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy idea. For this goal, classical effects are provided with new trouble-free proofs. then again, you will begin extra frequently with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, telephone complexes and fibre bundles. a unique function is the wealthy provide of approximately 500 workouts and difficulties. numerous sections contain themes that have no longer seemed ahead of in textbooks in addition to simplified proofs for a few vital effects. necessities are usual element set topology (as recalled within the first chapter), undemanding algebraic notions (modules, tensor product), and a few terminology from class conception. the purpose of the ebook is to introduce complex undergraduate and graduate (master's) scholars to simple instruments, innovations and result of algebraic topology. adequate history fabric from geometry and algebra is integrated. A book of the ecu Mathematical Society (EMS). disbursed in the Americas by way of the yank Mathematical Society.

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**Example text**

Z/ W X Z ! Y Z is a quotient map. Proof. We verify for p id the universal property of a quotient map: If h W Y Z ! p id/ is continuous, then h is continuous. p id/ is h^ ı p. 2), it is continuous. Since p is a quotient map, h^ is continuous. 3). 7) Theorem (Exponential law). Let X and Y be locally compact. Then the adjunction map ˛ W Z X Y ! Z Y /X is a homeomorphism. 4. Mapping Spaces and Homotopy 39 Proof. ˛ id/ is continuous. ˛1 id/ is continuous. One verifies that ˛2 D eX Y;Z . 1). eX;Z Y id/.

R X y/ has always two elements. 1 Path categories. Forming the product path is not an associative composition. We can remedy this defect by using parameter intervals of different length. So let us consider paths of the form u W Œ0; a ! X , v W Œ0; b ! 0/ and a; b 0. Their composition v ı u D w is the path Œ0; a C b ! a t / for a Ä t Ä a C b. X /: Objects are the points of X; a morphism from x to y is a path u W Œ0; a ! a/ D y for some a 0; and composition of morphisms is as defined before; the path Œ0; 0 !

2/ If X is a Hausdorff space, then X H is closed. 3/ Let A be a G-stable subset of the G-space X . Then A=G carries the subspace topology of X=G. In particular X G ! X ! X=G is an embedding. 4/ Let B X be closed and A X . Then fg 2 G j gA Bg is closed in G. 5/ Let B X be closed. Then fg 2 G j gB D Bg is closed. Proof. (1) The isotropy group Gx is the pre-image of x under the continuous map G ! X, g 7! gx. (2) The set X g D fx 2 X j gx D xg T is the pre-image of the diagonal under X ! X X , x 7!