Download e-book for kindle: A First Course in Algebraic Topology by A. Lahiri

By A. Lahiri

ISBN-10: 1842650033

ISBN-13: 9781842650035

This quantity is an introductory textual content the place the subject material has been provided lucidly on the way to support self examine by way of the newbies. New definitions are through appropriate illustrations and the proofs of the theorems are simply available to the readers. adequate variety of examples were integrated to facilitate transparent knowing of the techniques. The booklet begins with the fundamental notions of class, functors and homotopy of constant mappings together with relative homotopy. basic teams of circles and torus were handled in addition to the basic crew of overlaying areas. Simplexes and complexes are awarded intimately and homology theories-simplicial homology and singular homology were thought of in addition to calculations of a few homology teams. The ebook may be best suited to senior graduate and postgraduate scholars of assorted universities and institutes.

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Example text

Fundamental Group of the Circles 45 In this case we say that the path a is lifted to a path a' in R via 0 such that II y - y' II < 8 gives llf(y)-f(y') II< 1.

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.

Then F is a strong deformation retract of R11+1 - 0. 12. In R2 let C1 = {x =(x1> x2) : (x1 - 1)2 + x~ = 1} and = = and let Y C 1 u C2, X Y- {(2, 0), (-2, 0)}. We show that the point x0 = (0, 0) is a strong deformation retract of X. Let i: {x0 } ~ X and r: l{ ~ ·{x0 } be maps. Then ri =I. To see that ir :: I (rel {x0 }), we define the homotopy F:xxc~x = (1 - t) xi II ((1 - t) x 1 + (-l)k, (1 - t) x 2) II Ck> k l, 2. The F(x, t) is defined finitely because for x E X, ((i - t) x 1 + (-l)k, (1 - t) x2) (0, 0).

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A First Course in Algebraic Topology by A. Lahiri

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