Topology

A First Course in Algebraic Topology by A. Lahiri

By A. Lahiri

This quantity is an introductory textual content the place the subject material has been awarded lucidly as a way to aid self learn via the rookies. New definitions are via appropriate 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 innovations. The e-book begins with the fundamental notions of class, functors and homotopy of constant mappings together with relative homotopy. primary teams of circles and torus were handled in addition to the basic crew of masking areas. Simplexes and complexes are offered intimately and homology theories-simplicial homology and singular homology were thought of in addition to calculations of a few homology teams. The booklet can be best suited to senior graduate and postgraduate scholars of varied universities and institutes.

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However a local homeomorphism need not be a covering map as the following example shows. 7. Let Pt : (0, 3 )4 st be the restrictioQ of the map p : R 4 st defined by p(t) = e 2nit to the open interval (0, 3). Since p is a cov~ring map (see Ex. 2 and so its restriction Pt to the open set (0, 3) is a local homeomorphism. Pt is also a surjection. But since the complex number 1 e st has no neighbourhood evenly covered by Pt> Pt is not a covering map. Clearly a covering map is a surjection and so using the above lemmas, we obtain the following.

X be defined by H(u, v) = F(2u, v), 0 :S u :S 21 and H(u, v ) = 'j'(2u-1, v), 2I :Su :S 1. 3, H is continuous and H(u, 0) =f1(2u), 0 :Su :S H(u, 0) =Ji(2u - 1), 21 I 2' :Su :S 1, H(u, 1) 1 = g 1(2u), 0 :Su S 2' H(u, 1) = g2(2u - 1), 1 2 :Su :S 1, = F(O, v) =f1(0) =g1(0) H(l, v) ='j'(l, v) =Ji(l) =g2(1). H(O, v) and These relations show thatf1 *Ji - g 1 * g2 and this proves the theorem. 6. If f and g be paths such that f - g then J - g. Proof Sincef- g, there is a continuous mapping F: C x C ~ X such that F(u, 0) =f(u), F(u, 1) =g(u), F(O, v ) = f(O) = g(O), F(l, v) =f(l) = g(l).

3) is not true. A contractible space is of the same homotopy type as a space that contains only a single point. But these two spaces need not be homeomorphic (if the contractible space contains more than one point). We consider another example. S. B" is not homeomorphic to a single point, say {y0 } c B" but we show that B'1 is of the same homotopy type as a single point. e. f(y) = y) and the constant map g : Bn ~ {y}. Then gf =I and F : B" x C ~ B'1 defined by F(x, t) = tx + (1 - t)y is a homotopy between jg and I8 n.

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