# Algebraic topology by Tom Dieck T. 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 offers extra complex functions and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory path with homotopy concept. For this function, classical effects are offered with new trouble-free proofs. however, one can begin extra generally 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 offer of approximately 500 workouts and difficulties. numerous sections comprise issues 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), effortless algebraic notions (modules, tensor product), and a few terminology from type concept. the purpose of the e-book is to introduce complicated undergraduate and graduate (master's) scholars to simple instruments, suggestions and result of algebraic topology. adequate historical past fabric from geometry and algebra is integrated. A e-book of the eu Mathematical Society (EMS). dispensed in the Americas by means of the yankee Mathematical Society.

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X/k < 2. This indicates another use of homotopies: Improve maps up to homotopy. If one uses some analysis, namely (the easy part of) the theorem of Sard about the density of regular values, one sees that for m < n a C 1 -map S m ! S n is not surjective and hence null homotopic. ) There exist surjective continuous maps S 1 ! 4) Proposition. The map p W S n 1 I ! x; t / 7! 1 t /x is a quotient map. Given F W D n ! X , the composition Fp W S n 1 I ! X is a null homotopy of f D F jS n 1 . Each null homotopy of a map f W S n 1 !

Thus X is compact if and only if X=G is compact. (4) If G is compact and X separated, then X=G is separated. (5) Let G be compact, A a G-stable closed subset and U a neighbourhood of A in X. Then U contains a G-stable neighbourhood of A. Proof. (1) A B G X is compact as a product of compact spaces. Hence the continuous image AB of A B under r W G X ! X is compact. (2) The homeomorphism A X ! s; x/ 7! s; sx/ transforms r into the projection pr W A X ! X . 1)). Hence the image AB of the closed set A B is closed.

Z, and gf are h-equivalences, then so isQ the third. Homotopy is compatible with sums and products. Let pi W j 2J Xj ! Xi be the projection onto the i -th factor. Then Q Q ŒY; j 2J Xj  ! j 2J ŒY; Xj ; Œf  7! Œpi ı f / ` is a well-defined bijection. Let ik W Xk ! j 2J Xj be the canonical inclusion of the k-th summand. Then Q ` Œ j 2J Xj ; Y  ! j 2J ŒXj ; Y ; Œf  7! Œf ı ik / is a well-defined bijection. In other words: sum and product in TOP also represent sum and product in h-TOP. ) Let P be a point.
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