By Czes Kosniowski

ISBN-10: 0521298644

ISBN-13: 9780521298643

This self-contained advent to algebraic topology is appropriate for a few topology classes. It includes approximately one zone 'general topology' (without its traditional pathologies) and 3 quarters 'algebraic topology' (centred round the primary staff, a with ease grasped subject which provides a good suggestion of what algebraic topology is). The ebook has emerged from classes given on the collage of Newcastle-upon-Tyne to senior undergraduates and starting postgraduates. it's been written at a degree in order to allow the reader to take advantage of it for self-study in addition to a direction e-book. The strategy is leisurely and a geometrical flavour is obvious all through. the various illustrations and over 350 routines will end up worthy as a educating relief. This account may be welcomed by means of complicated scholars of natural arithmetic at schools and universities.

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**Example text**

10 Definition Suppose that X is a topological space and G is a group then we say that X is a G-space if G acts on X and if the function 0g given by x -+ g E G. 11 Exercise Suppose that x is a G-space. Prove that the function 0g given by x -, is a homeomorphism from X to itself for all g E G. Deduce that there is a homomorphism from G to the group of homeomorphisms of X. Because of the above exercise we sometimes say that if X is a G-space then G is a group of homeomorphlsms of X. Using this we prove the next result.

What if we cut along a line one-third of the distance between the edges? The next result gives sufficient conditions to ensure that quotients of homeomorphic spaces are homeomorphic. 5 Theorem Let f: X —* Y be a function between the topological spaces X and Y. Suppose that X and Y have equivalence relations respectively and such that x if and only if f(x) f(x'). If f is a homeomorphism then and are homeomorphic. Proof Define a function F: -+ by F [x] = [f(x)], where the square brackets denote equivalence classes.

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'). 5 the space Regluing Y according to Z is homeomorphic to the space X. A concept that we shall find useful later on is that of a group G 'acting' on a set X. The notion is fruitful and leads to examples of spaces with the quotient topology.

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

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