By A. Lahiri

This quantity is an introductory textual content the place the subject material has been awarded lucidly as a way to support self research through the newcomers. New definitions are via compatible illustrations and the proofs of the theorems are simply obtainable to the readers. enough variety of examples were included to facilitate transparent knowing of the techniques. The ebook begins with the fundamental notions of class, functors and homotopy of continuing mappings together with relative homotopy. primary teams of circles and torus were taken care of besides the elemental crew of protecting areas. Simplexes and complexes are provided intimately and homology theories-simplicial homology and singular homology were thought of besides calculations of a few homology teams. The e-book could be best suited to senior graduate and postgraduate scholars of varied universities and institutes.

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8. Covering Spaces _ _ __ 1. Definitions The idea of covering spaces is very important in topological spaces which found applications in related disciplines such as differential geometry, the theory of Lie groups and the theory of Riemann surfaces. This idea is also closely connected with the study of fundamental group. Several topological problems about covering spaces can be reduced to algebraic problems on the fundamental groups of the concerned spaces. Below we begin with the definition of a covering space ..

Definitions Let X be a topological space and C denotes the unit interval [O, 1]. A path in Xis a continuous mapping/: C ~ X. The points/(O) and/(1) are called respectively the initial point and the terminal point of the path. It should be noted that the path is the mapping f and not the image /([0, 1]). If x is a fixed element of X, we shall denote by Ex the constant mapping Ex: [0, l] ~ X = given by Ex(t) x for every t e C. The path Ex is called a null path. 1. Ex is a path. The inverse path 1 off is defined to be the path such that 1 (t) =JO - t), o s t s 1.

3. Let X be connected and has the same homotopy type as Y. Show that Y is connected. 4. Let A c B c X. Suppose that B is a retract of X and A is a retract of B. Show that A is a retract of X. 5. Let X be locally compact and A be a retract of X. Prove that A is also locally compact. 6. Show that a retract of a Hausdorff space is a closed subset. 7. Show that {O} u { 1} is not a retract of C. 8. Prove that there is a retraction r : B" --+ s•1-I if and only if sn-I is contractible. (Hint : Let F : s11- 1 x c--+ s11- 1 be a homotopy between a constant map and the identity map.