By Tammo Tom Dieck
This ebook is written as a textbook on algebraic topology. the 1st half covers the fabric 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 path with homotopy conception. For this goal, classical effects are offered with new easy proofs. on the other hand, you may commence extra generally 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 offer of approximately 500 workouts and difficulties. numerous sections contain subject matters that have now not seemed prior to in textbooks in addition to simplified proofs for a few very important effects. necessities are usual aspect set topology (as recalled within the first chapter), common algebraic notions (modules, tensor product), and a few terminology from classification conception. the purpose of the booklet is to introduce complicated undergraduate and graduate (master's) scholars to easy instruments, suggestions and result of algebraic topology. enough historical past fabric from geometry and algebra is integrated. A booklet of the ecu Mathematical Society (EMS). dispensed in the Americas via the yankee Mathematical Society.
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Extra info for Algebraic Topology (EMS Textbooks in Mathematics)
V on a real (or complex) vector space V is called a real (or complex) representation of G if the left translations are linear maps. C/. A homomorphism G ! n/ or G ! n/ is called an orthogonal or unitary representation. Geometrically, an orthogonal representation is given by an action G V ! V with an invariant scalar product h ; i. The latter means hgv; gwi D hv; wi for g 2 G and v; w 2 V . 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/.
As in any category we also have the Hom-functors in h-TOP. Given f W X ! Y , we use the notation f W ŒZ; X ! ŒZ; Y ; g 7! fg; f W ŒY; Z ! ŒX; Z; h 7! 1. The Notion of Homotopy 29 for the induced maps2 . The reader should recall a little reasoning with Homfunctors, as follows. , an isomorphism in h-TOP if and only if f is always bijective; similarly for f . If f W X ! Y has a right homotopy inverse h W Y ! , f h ' id, and a left homotopy inverse g W Y ! , gf ' id, then f is an h-equivalence.
C / A X . C / D A \ U with an open subset U have A \ U D A \ GU , since A is G-stable. GU /. GU / is open, hence C is open in the subspace topology. By continuity of A=G ! 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 !
Algebraic Topology (EMS Textbooks in Mathematics) by Tammo Tom Dieck