By Tammo Tom Dieck

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 conception. For this function, classical effects are awarded with new simple proofs. however, 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 different function is the wealthy offer 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 typical element set topology (as recalled within the first chapter), simple algebraic notions (modules, tensor product), and a few terminology from class conception. the purpose of the booklet is to introduce complicated undergraduate and graduate (master's) scholars to uncomplicated instruments, thoughts and result of algebraic topology. enough historical past fabric from geometry and algebra is incorporated. A ebook of the ecu Mathematical Society (EMS). allotted in the Americas by way of the yankee Mathematical Society.

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**Example text**

Y; Z// ! X ^ Y; Z/. 0 be continuous. Then ˛ 0 and ˇ 0 are inverse bijections. 9) Proposition. 10) Corollary. X ^ Y / I ! Z be a pointed homotopy. h t / W X ! Y; Z/ is a pointed homotopy and therefore ŒX ^ Y; Z0 ! Y; Z/0 ; Œf 7! f / 0 is continuous, then this map is bijective. is well defined. 7), we obtain the pointed version of the exponential law. 11) Theorem (Exponential law). Let X and Y be locally compact. X ^ Y; Z/ ! Y; Z// is a homeomorphism. 12) Lemma. Let ka W Z ! A denote the constant map with value a.

X / is called the fundamental groupoid of X . The automorphism group of the object x in this category is the fundamental group of X with respect to the base point x. The usual rules of categorical notation force us to define the multiplication in this group by Œu ı Œv D Œv u. X; x/. X; x/ (Poincaré 1895 [151, §12]). 1/ D x), also called a loop based at x. 2) Remark. X /. In that case we call paths u W Œ0; a W I ! X and v W Œ0; b ! 5. The Fundamental Groupoid 43 such that the compositions with u and v, respectively, have the same domain of definition Œ0; c and the resulting composed paths are homotopic rel f0; cg.

T " ..... ... . x . 0 H1 D v Ht H0 D u .... x1 .... .... H constant along dotted lines .... ... x1 42 Chapter 2. The Fundamental Group Being homotopic in this sense is an equivalence relation on the set of all paths from x to y. The product operation is compatible with this relation as the next proposition shows. 1) Proposition. The product of paths has the following properties: (1) Let ˛ W I ! 1/ D 1. Then u ' u˛. u1 u2 / u3 (if the products are defined ). (3) u1 ' u01 and u2 ' u02 implies u1 u2 ' u01 u02 .

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