Algebraic Geometry

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By Tom Dieck T.

This e-book 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 offers extra complex purposes 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 basic proofs. however, one can begin extra often with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, mobile complexes and fibre bundles. a different function is the wealthy provide of approximately 500 routines and difficulties. a number of sections contain themes that have now not seemed sooner than in textbooks in addition to simplified proofs for a few very important effects. necessities are regular element set topology (as recalled within the first chapter), ordinary algebraic notions (modules, tensor product), and a few terminology from classification conception. the purpose of the ebook is to introduce complicated undergraduate and graduate (master's) scholars to easy instruments, techniques and result of algebraic topology. adequate historical past fabric from geometry and algebra is incorporated. A ebook of the ecu Mathematical Society (EMS). dispensed in the Americas via the yank Mathematical Society.

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Example text

Xj ; Y /, since W 0 j Xj is the sum in TOP . If J is finite, it is a homeomorphism. 7. Let X; Y; U , and V be spaces. The Cartesian product of maps gives a map W UX V Y ! f; g/ 7! f g: Let X and Y be Hausdorff spaces. Then the map is continuous. 8. By definition of a product, a map X ! 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 !

X; 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. If two of the maps f W X ! Y , g W Y ! Z, and gf are h-equivalences, then so isQ the third. Homotopy is compatible with sums and products. Let pi W j 2J Xj !

It is a homeomorphism if X Y ! , if X and Y =B are locally compact (or in the category of compactly generated spaces). 3 Standard Spaces Standard spaces are Euclidean spaces, disks, cells, spheres, cubes and simplices. We collect notation and elementary results about such spaces. The material will be used almost everywhere in this book. We begin with a list of spaces. The Euclidean norm is kxk. n 1/-dimensional sphere n-dimensional cell n-dimensional cube boundary of I n n-dimensional simplex boundary of n The spaces D n , I n , E n and n are convex and hence contractible.

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