By A. Lahiri

This quantity is an introductory textual content the place the subject material has been provided lucidly to be able to aid self research through the rookies. New definitions are by way of compatible illustrations and the proofs of the theorems are simply obtainable to the readers. adequate variety of examples were integrated to facilitate transparent realizing of the suggestions. The publication begins with the elemental notions of type, functors and homotopy of continuing mappings together with relative homotopy. primary teams of circles and torus were taken care of in addition to the elemental staff of overlaying areas. Simplexes and complexes are awarded intimately and homology theories-simplicial homology and singular homology were thought of besides calculations of a few homology teams. The ebook might be best suited to senior graduate and postgraduate scholars of varied universities and institutes.

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**Sample text**

On the other hand, the following path at 1 y(t) =e 211i1, t E C makes exactly one revolution of st. This map is not homotopic to the path Et and so belongs to a nontrivial homotopy class. There may be other type of path at 1, those make one complete revolution and without stoping at 1 these go further to a point e;90 (without making a second complete revolution) and then retraces-back to the point 1. e. homotopic to Et), so that such paths are considered to produce exactly one revolution of st.

A. E A} is a collection of path connected subsets of X such that Then y = u YA. is path connected. eA Proof. · Since Yk is path connected and p, r E Ykt there is a path f from p to r. Similarly, there is a path g from r to q. Let h = f * g be defined by ! (2t), 0::;; t::;; h(t) ={ 1 g (2t - 1), 2 t ::;; t ::;; 1. Then his a path from p to q. This proves the theorem. 2. If n ~ 1 then R11+ 1 - {O} is path connected. 3. S" is path connected (n ~ 1). 4. Every path connected space is connected. Proof.

Let = Proof Let Y denote either C or C x C,f: Y ~ S 1 denote either a or F and 0 e Y means either 0 e R or (0, 0) e C x C. Because Y is compact, f is uniformly continuous on Y. So, there exists 8 > 0 such that II y - y' II < 8 gives llf(y)-f(y') II< 1. e. f(y)/f(y') * -1. Therefore lfl(f(y)lf(y')) is defined whenever II y - y' II < 8. Let N be a positive integer such that II y II