By Czes Kosniowski

ISBN-10: 0521298644

ISBN-13: 9780521298643

This self-contained creation to algebraic topology is acceptable for a few topology classes. It includes approximately one zone 'general topology' (without its ordinary pathologies) and 3 quarters 'algebraic topology' (centred round the basic crew, a effortlessly grasped subject which provides a good suggestion of what algebraic topology is). The e-book has emerged from classes given on the college of Newcastle-upon-Tyne to senior undergraduates and starting postgraduates. it's been written at a degree on the way to permit the reader to take advantage of it for self-study in addition to a direction booklet. The method is leisurely and a geometrical flavour is obvious all through. the various illustrations and over 350 routines will turn out necessary as a educating reduction. This account might be welcomed by means of complicated scholars of natural arithmetic at schools and universities.

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**Extra resources for A First Course in Algebraic Topology**

**Sample text**

Show that { ghhEU} ,gEG,UE5°(G) defines an action of G on5'°(G). (d) Let G act on X and define the stabilizer of x E X to be the set {gEG;gx=x}. Prove that is a subgroup of G. (e) Let G act on X and define the orbit of x E X to be the subset 1 gx;gEG } of X. Prove that two orbits Gx, are either disjoint or equal. Deduce that a Gset X decomposes into a union of disjoint subsets. An important consequence of the definition of a G-set X is that in fact G acts on X via bijections. 8 Theorem x -+ is bijective.

The function f: R + -* R given by f(x) = log3(x) is a f(x) f(x'), hence the spaces R + /'- and homeomorphism and x ' x' R /- are homeomorphic; in fact both are homeomorphic to the circle. 5 explains the intuitive idea of a 'homeomorphism' as presented in Chapter 4: We start with a space W. By cutting it we get X and a relation which tells us how to reglue X in order to get W. Now perform a homeomorphism f on X to give Y with an equivalence relation NaturAs an example consider x ally, we want that x x' • f(x) y f(x').

Thus f is continuous. Next, suppose that W is an open subset of X X y I so that W = (U X n X X { y } where are open in X,Y respecJ tively. W may be rewritten as U X { y } where J' { j E J; y E } JEJ' thus f(W) = U which is open in X. This proves that f is also open and JEJ' hence is a homeomorphism. If' f: A -÷ X and g: A Y are mappings between topological spaces then we can define a mapping h: A -+ X X Y by h(a) = (f(a),g(a)). It is clear that h is the unique mapping such that lrxh = f and iryh = g.

### A First Course in Algebraic Topology by Czes Kosniowski

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