By A. Lahiri

This quantity is an introductory textual content the place the subject material has been provided lucidly with the intention to aid self research by means of 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 integrated to facilitate transparent realizing of the innovations. The publication begins with the fundamental notions of classification, functors and homotopy of continuing mappings together with relative homotopy. basic teams of circles and torus were taken care of besides the basic team 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 publication may be best suited to senior graduate and postgraduate scholars of assorted universities and institutes.

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S11 is a strong deformation retract of R11+1 - 0. Let F : (R 11+1 - 0) x C ~ R11+1 - 0 be defined by F(x, t) = (1 - t) x + II~ II' x e R11+1 - 0, t e C. 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.

50 ALGEBRAIC TOPOLOGY Since G is continuous, we find that lim F ( t, s) = lim G ( t, s/l - s) = lim G ( t, r) s-71 s-71 r-700 = (exp (21tit))k and so Fis also continuous. Also we verify that Fis a homotopy (rel A) between f 0(t) = F(t, 0) and f 1(t) = F(t, 1). Hence, fo - f 1 and so by our preceding observation deg

Further =F(f(x), 0) =f(x) ;f (x, 1) = F(j(x), 1) =Yo· ;f (x, 0) and This shows that f is homotopic to g : X ~ Y where g is a constant mapping defined by g(x) =y0 for every x e X. This proves the theorem. 1. It follows from this theorem that in a contractible space there is only one homotopy class of mappings. 2. Homotopy Type A space X is said to be of the same homotopy type as Y if there exist continuous mappings f : X ~ Y and g : Y ~ X such that gf : X ~ X is homotopic to Ix andfg: Y ~ Y is homotopic to ly.

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