By Tom Dieck T.

ISBN-10: 3037190485

ISBN-13: 9783037190487

This publication 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 complicated purposes and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy thought. For this goal, classical effects are offered with new simple proofs. however, it is easy to begin extra typically with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, phone complexes and fibre bundles. a unique characteristic is the wealthy offer of approximately 500 workouts and difficulties. a number of sections contain themes that have now not seemed earlier than in textbooks in addition to simplified proofs for a few vital effects. necessities are typical element set topology (as recalled within the first chapter), basic algebraic notions (modules, tensor product), and a few terminology from classification concept. the purpose of the ebook is to introduce complex undergraduate and graduate (master's) scholars to easy instruments, thoughts and result of algebraic topology. enough history fabric from geometry and algebra is integrated. A booklet of the eu Mathematical Society (EMS). dispensed in the Americas by way of the yankee Mathematical Society.

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**Example text**

Y . A space is contractible if it is homotopy equivalent to a point. A map f W X ! Y is null homotopic if it is homotopic to a constant map; a null homotopy of f is a homotopy between f and a constant map. 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 !

1. The Notion of Homotopy 27 A space is connected if it is not the topological sum of two non-empty subspaces. Thus X is disconnected if and only if X contains a subset X which is open, closed, and different from ; and X. A decomposition of X is a pair U; V of open, nonempty, disjoint subsets with union X . A space X is disconnected if and only if there exists a continuous surjective map f W X ! 1/. The continuous image of a connected space is connected. Recall from calculus: A R is connected if and only if A is an interval.

X . homotopies H correspond via H The map p induces a homeomorphism px W S n 1 I =S n 1 1 ! 3)). x. Hence there exists a unique F such that F px D H 36 Chapter 2. n/ D Rn [ f1g; since these spaces are homeomorphic to S n . The canonical map I nCm =@I nCm ! I m @I m ^ I n =@I n which is the identity on representatives is a pointed homeomorphism. If V and W are finite-dimensional real vector spaces, we have a canonical pointed homeomorphism S V ^ S W Š S V ˚W which is the identity away from the 2s 1 base point.

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