# Algebraic Topology (EMS Textbooks in Mathematics) by Tammo Tom Dieck

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 offers extra complicated functions and ideas (duality, attribute periods, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy concept. For this goal, classical effects are awarded with new undemanding proofs. however, you'll be able to commence extra normally with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, mobilephone complexes and fibre bundles. a different function is the wealthy provide of approximately 500 routines and difficulties. a number of sections comprise issues that have no longer 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), uncomplicated algebraic notions (modules, tensor product), and a few terminology from type idea. the purpose of the booklet is to introduce complex undergraduate and graduate (master's) scholars to simple instruments, thoughts and result of algebraic topology. adequate heritage fabric from geometry and algebra is incorporated. A booklet of the ecu Mathematical Society (EMS). disbursed in the Americas through the yank Mathematical Society.

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Let p W X ! Y be a surjective continuous map. Suppose the pre-image of a compact set is compact. Then Z p W Z Y ! Z X is an embedding. W Q 6. We have a canonical bijective map F 0 j 2J Xj ; Y ! Xj ; Y /, since W 0 j Xj is the sum in TOP . If J is finite, it is a homeomorphism. 7. Let X; Y; U , and V be spaces. The Cartesian product of maps gives a map W UX V Y ! f; g/ 7! f g: Let X and Y be Hausdorff spaces. Then the map is continuous. 8. By definition of a product, a map X ! Y Z is essentially the same thing as a pair of maps X !

3) u1 ' u01 and u2 ' u02 implies u1 u2 ' u01 u02 . (4) u u is always defined and homotopic to the constant path. 1/ . Proof. s; t / 7! s// is a homotopy from u to u˛. t / D 2t C 12 for 12 Ä t Ä 1. 2s 1; t / for 12 Ä t Ä 1. (4) The map F W I I ! 1 t / for 12 Ä t Ä 1 is a homotopy from u u to the constant path. ) (5) is proved again with the parameter invariance (1). X/. The objects are the points of X . A morphism from x to y is a homotopy class relative to @I of paths from x to y. A constant path represents an identity.

Proof. Since a null homotopy H of f W S n 1 ! X sends S n 1 1 to a point, it factors through the quotient map q W S n 1 I ! S n 1 I =S n 1 1. Thus null x 7! H x q to maps H x W S n 1 I =S n 1 1 ! X . homotopies H correspond via H The map p induces a homeomorphism px W S n 1 I =S n 1 1 ! 3)). x. Hence there exists a unique F such that F px D H 36 Chapter 2. n/ D Rn [ f1g; since these spaces are homeomorphic to S n . The canonical map I nCm =@I nCm ! I m @I m ^ I n =@I n which is the identity on representatives is a pointed homeomorphism.