By Tom Dieck T.

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 provides extra complicated functions and ideas (duality, attribute sessions, homotopy teams of spheres, bordism). the writer recommends beginning an introductory direction with homotopy conception. For this goal, classical effects are provided with new basic proofs. then again, it is easy to commence extra usually with singular and axiomatic homology. extra chapters are dedicated to the geometry of manifolds, mobile complexes and fibre bundles. a unique characteristic is the wealthy provide of approximately 500 workouts and difficulties. numerous sections contain issues that have no longer seemed sooner than in textbooks in addition to simplified proofs for a few vital effects. necessities are average aspect set topology (as recalled within the first chapter), effortless algebraic notions (modules, tensor product), and a few terminology from class conception. the purpose of the booklet is to introduce complicated undergraduate and graduate (master's) scholars to easy instruments, strategies and result of algebraic topology. adequate history fabric from geometry and algebra is integrated. A booklet of the ecu Mathematical Society (EMS). disbursed in the Americas through the yank Mathematical Society.

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**Example text**

Aj j j 2 JS/ be a family of connected subsets of X such that Ai \ Aj 6D ; for all i; j . Then j Aj D Y is connected. Let A be connected x Then B is connected. and A B A. The union of the connected sets in X which contain x is thus a closed connected subset. x/ of x in X . x/. A component of X is a maximal connected subset. Any space is the disjoint union of its components. A space is totally disconnected if its components consist of single points. Since intervals are connected a path connected space is connected.

X/ is a homotopy from f to g (linear homotopy). It will turn out that many homotopies are constructed from linear homotopies. A set A Rn is star-shaped with respect to a0 2 A if for each a 2 A the line-segment from a0 to a is contained in A. 1 t/a C ta0 is a null homotopy of the identity. Hence star-shaped sets are contractible. A set C Rn is convex if and only if it is star-shaped with respect to each of its points. Note: If A D Rn and a0 D 0, then each H t , t < 1, is a homeomorphism, and only in the very last moment is H1 constant!

One idea of deformation is made precise by the notion of homotopy. Continuity is an ungeometric notion. So often one has to deform a continuous map into a map with better properties. (3) The homotopy notion leads in an almost tautological way to algebraic structures and categorical structures. In this chapter we learn about the simplest example, the fundamental group and the fundamental groupoid. The passage to the homotopy category is not always a suitable view-point. In general it is better to stay in the category TOP of topological spaces and continuous maps (“space level” as opposed to “homotopy level”).