By Tom Dieck T.

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 complicated functions and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy conception. For this objective, classical effects are offered with new undemanding proofs. on the other hand, you'll commence extra usually with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, cellphone complexes and fibre bundles. a different function is the wealthy provide of approximately 500 routines and difficulties. numerous sections comprise subject matters that have now not seemed earlier than in textbooks in addition to simplified proofs for a few very important effects. necessities are commonplace element set topology (as recalled within the first chapter), effortless algebraic notions (modules, tensor product), and a few terminology from classification thought. the purpose of the publication is to introduce complex undergraduate and graduate (master's) scholars to uncomplicated instruments, strategies and result of algebraic topology. enough historical past fabric from geometry and algebra is incorporated. A booklet of the ecu Mathematical Society (EMS). allotted in the Americas by way of the yankee Mathematical Society.

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**Example text**

D mCn ; S mCn 1 /. 2. RnC1 X fxg ! S n , z 7! z x/=kz xk is an h-equivalence. n n 3. x0 ; : : : ; xn / 2 S n j xn 0g. Show that the quotient map S n ! S n =DC is an h-equivalence. 4. Let f1 ; : : : ; fk W Cn ! C be linearly independent linear forms (k Ä n). 0/ is homotopy equivalent to the product of k factors S 1 . 5. S n ! x; y/ 2 S n S n j x 6D yg, x 7! x; x/ is an h-equivalence. 6. Let f; g W X ! x/. Then f ' g. 7. Let A E n be star-shaped with respect to 0. Show that S n 1 Rn XA is a deformation retract.

5. 7. X / is null homotopic. 8. gf is null homotopic, if f or g is null homotopic. 9. Let A be contractible. Then any two maps X ! A are homotopic. 10. C/ are homotopy equivalences. n; n/-matrices. n/ ! X; P / 7! n/ is star-like with respect to the unit matrix. 11. There exist contractible and non-contractible spaces consisting of two points. 2 Further Homotopy Notions The homotopy notion can be adapted to a variety of other contexts and categories: Consider homotopies which preserve some additional structure of a category.

N k/ 0 B 0 B respectively. The map A 7! Rn /. 9. Projective Spaces. Rn / is a compact Hausdorff space. It is called the Grassmann manifold of k-dimensional subspaces of Rn . Cn /. Chapter 2 The Fundamental Group In this chapter we introduce the homotopy notion and the first of a series of algebraic invariants associated to a topological space: the fundamental group. Almost every topic of algebraic topology uses the homotopy notion. Therefore it is necessary to begin with this notion. A homotopy is a continuous family h t W X !