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 purposes and ideas (duality, attribute periods, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy conception. For this function, classical effects are offered with new ordinary proofs. then again, you can still commence extra regularly with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, telephone complexes and fibre bundles. a unique characteristic is the wealthy offer of approximately 500 routines and difficulties. a number of sections comprise subject matters that have now not seemed prior to in textbooks in addition to simplified proofs for a few vital effects. must haves are regular aspect set topology (as recalled within the first chapter), common algebraic notions (modules, tensor product), and a few terminology from classification idea. the purpose of the publication is to introduce complicated undergraduate and graduate (master's) scholars to simple instruments, innovations and result of algebraic topology. adequate heritage fabric from geometry and algebra is integrated. A e-book of the eu Mathematical Society (EMS). disbursed in the Americas by means of the yank Mathematical Society.
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Additional info for Algebraic Topology (EMS Textbooks in Mathematics)
5/ Let B X be closed. Then fg 2 G j gB D Bg is closed. Proof. (1) The isotropy group Gx is the pre-image of x under the continuous map G ! X, g 7! gx. (2) The set X g D fx 2 X j gx D xg T is the pre-image of the diagonal under X ! X X , x 7! x; gx/, and X H D g2H X g . (3) Let C A=G be open with respect to the quotient map A ! A=G. 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 !
Y Z is essentially the same thing as a pair of maps X ! Y , X ! Z. Y Z/X ! Y X Z X . Let X be a Hausdorff space. Then the tautological map is a homeomorphism. 9. Let X and Y be locally compact. Then composition of maps Z Y Y X ! g; f / 7! g ı f is continuous. 10. X; A/ a pair of spaces and p W X ! X=A the quotient map. The space X=A is pointed with base point fAg. X; Y / of the maps which send A to the base point. X=A; Y / ! 5. The Fundamental Groupoid 41 ŒX=A; Y 0 ! Y; /. If p has compact pre-images of compact sets, then is a homeomorphism.
X/ is a contraction of the space X . 4 Categories of homotopies. X; Y /. The objects are the continuous maps f W X ! Y . A morphism from f to g is a homotopy H W X Œ0; a ! Y with H0 D f and Ha D g. 1). Þ 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 .