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 complicated purposes and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy thought. For this function, classical effects are awarded with new effortless proofs. on the other hand, you possibly can commence extra regularly with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, cellphone complexes and fibre bundles. a distinct function is the wealthy offer of approximately 500 routines and difficulties. a number of sections contain subject matters that have now not seemed ahead of in textbooks in addition to simplified proofs for a few vital effects. necessities are general aspect set topology (as recalled within the first chapter), hassle-free algebraic notions (modules, tensor product), and a few terminology from class conception. the purpose of the publication is to introduce complicated undergraduate and graduate (master's) scholars to simple instruments, ideas and result of algebraic topology. enough history fabric from geometry and algebra is integrated. A book of the eu Mathematical Society (EMS). dispensed in the Americas by means of the yank Mathematical Society.

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V / D fv 2 V j hv; vi D 1g is G-stable. Let E be a right G-space and F a left G-space. E F / ! x; y// 7! xg 1 ; gy/. A G-map f W F1 ! F2 induces a continuous map id Gf WE G F1 ! x; y/ 7! x; y// 7! kx; y/. This construction can in particular be applied in the case that E D K, G a subgroup 20 Chapter 1. Topological Spaces of K and the G- and K-actions on K are given by right and left multiplication. The assignments F 7! K G F and f 7! id G f yield the induction functor indK G W G- TOP ! K- TOP. This functor is left adjoint to the restriction functor resK G W K- TOP !

X=G, an open subset in the subspace topology is open in A=G. (4) ra W G ! X ,Tg 7! B/ D fg 2 G j gA Bg closed. (5) The set fg j gB D Bg D fg j gB Bg \ fg j g 1 B Bg is closed, by (4). 5) Proposition. Let r W G X ! X be a G-action, A G and B X. 8. Transformation Groups 19 (1) If A and B are compact, then AB is compact. (2) If A is compact, then the restriction A X ! X of r is proper. If, moreover, B is closed, then AB is closed. (3) If G is compact, then the orbit map p is proper. Thus X is compact if and only if X=G is compact.

Transformation Groups 17 x is also. subgroup, then the closure of H is also. 8 Transformation Groups A left action of a topological group G on a topological space X is a continuous map W G X ! g; x/ 7! gh/x and ex D x for g; h 2 G, e 2 G the unit, and x 2 X . X; / consists of a space X and a left action of G on X . The homeomorphism lg W X ! X , x 7! gx is called left translation by g. We also use right actions X G ! x; g/ 7! hg/ and xe D x. For A X and K G we let KA D fka j k 2 K; a 2 Ag. An action is effective if gx D x for all x 2 X implies g D e.