Download PDF by A. Lahiri: A First Course in Algebraic Topology

By A. Lahiri

ISBN-10: 1842650033

ISBN-13: 9781842650035

This quantity is an introductory textual content the place the subject material has been awarded lucidly which will aid self examine by means of the rookies. New definitions are through compatible illustrations and the proofs of the theorems are simply obtainable to the readers. enough variety of examples were integrated to facilitate transparent figuring out of the innovations. The booklet starts off with the fundamental notions of type, functors and homotopy of constant mappings together with relative homotopy. basic teams of circles and torus were taken care of besides the elemental crew of overlaying areas. Simplexes and complexes are offered intimately and homology theories-simplicial homology and singular homology were thought of besides calculations of a few homology teams. The e-book might be best suited to senior graduate and postgraduate scholars of varied universities and institutes.

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1 + v)l(l + v)), (1 - v)/2:::; u:::; 1. u). It is now a matter of simple verification that Fis a homotopy relative to the set {0, 1}. This proves the theorem. 7 we know that if/ andg be paths then/- g if and only if f - g. The next theorems show the behaviour of the product ·paths of /and f. 11. If /is any path then null paths. Proof We define F : C x C F(u, v) and ~ f * f and f *f are homotopic to X by =f(2u(l - t t ~ v)), 0 $ u F(u, v) =/(2(1- u) (1 - v)), $ u $ 1. Clearly F is continuous. Moreover F(u, 0) =f(2u), 0 $ u $ F(u, 0) =f(2-2u), F(u, 1) t t $ u $ l =f(O) F(O, v) =f(O) =F(l, v)~ This gives that f * f is homotopic to the null path whose image isf(O).

13, if* g) * h- f* (g * h). So, the operation is associative. e. 9 [f] [/] = [/) = [/] [/]. ·33 34 ALGEBRAIC TOPOLOGY So, [/] is a unit-element (identity). 11. [fJ [f1 =U* 11 =[I] so that every element has an inverse. Thus n1(X, x 0 ) is a group. This proves the theorem. 1. n 1(X, x0 ) is called the fundamental group of X at x0 • From the structure of n1(X, x0) we may obtain local character of the space at x0 . e. the group whose only element is the identity [/], (I= e xo), then every path at x0 is homotopic to I and intuitively this means, for example, that there is no "holes" which prevent a path at x 0 from shrinking to the point x 0 • In the above, if instead of a closed path, arbitrary path is taken, there will be a problem of the idehtity and also the multiplication need not always be defined.

If f is a local homeomorphism then each point of Y has neighbourhood with the above property. 1. A local homeomorphism is an open map. 2. A covering map is a local homeomorphism. Proof. Let p : X ~ X be a covering map and let i E X. Let Ube an open neighbourhood of p (i) which is evenly covered by p. From definition p-1 (U)=\,JU· JEJ J with Ui n Uk =

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