By Tom Dieck T.

This publication 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 complex functions and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory path with homotopy idea. For this objective, classical effects are provided with new common proofs. on the other hand, you may begin extra typically with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, phone complexes and fibre bundles. a distinct function is the wealthy provide of approximately 500 workouts and difficulties. numerous sections contain themes that have now not seemed earlier than in textbooks in addition to simplified proofs for a few very important effects. must haves are ordinary element set topology (as recalled within the first chapter), straight forward algebraic notions (modules, tensor product), and a few terminology from class thought. the purpose of the ebook is to introduce complex undergraduate and graduate (master's) scholars to simple instruments, thoughts and result of algebraic topology. adequate historical past fabric from geometry and algebra is integrated. A book of the eu Mathematical Society (EMS). allotted in the Americas by way of the yank Mathematical Society.

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**Example text**

I; @I / ! 2/ although the component maps I ! I and @I ! I X f1=2g are h-equivalences. 8. Let E R2 consist of k points. Show, heuristically, that the complement R2 X E is W 1 h-equivalent to the k-fold sum k 1 S . 9. Remove a point from the torus S 1 S 1 and show that the result is h-equivalent to S 1 _S 1 . Is there an analogous result when you remove a point from S m S n ? 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 !

A space is totally disconnected if its components consist of single points. Since intervals are connected a path connected space is connected. A product …j Xj is connected if each Xj is connected. xj / 2 …j Xj is the product of the components of the xj . 3) Example. x; sin. x 1 / j 0 < x Ä 1g is connected but not path connected. The union S of X with f˙1g Œ 2; 0 [ Œ 2; 2 f 2g is called the pseudo-circle. The complement R2 X S has two path components. of compact neighbourhoods AT pseudo-circle S has a sequence K1 K2 Þ with i Ki D S and Ki homeomorphic to S 1 Œ0; 1.

G/. Proof. The claim says that for each path u W I ! 0/ gu holds. We use I I ! s; t / 7! s/; t /. t; 1/.