Ivancevic_Applied-Diff-Geom

494 Applied Differential Geometry: A Modern IntroductionConsider the short exact sequence of vector bundles over X,j0 → Y ′ i−→ Y −→Y ′′ → 0, (4.10)which means that i is a bundle monomorphism, j is a bundle epimorphism,and Ker j = Im i. Then Y ′′ is the factor bundle Y/Y ′ . One says thatthe short exact sequence (4.10) admits a splitting if there exists a bundlemonomorphism s : Y ′′ → Y such that j ◦ s = Id Y ′′, i.e.,Y = i(Y ′ ) ⊕ s(Y ′′ ) ∼ = Y ′ ⊕ Y ′′ .Vector bundles of rank 1 are called line bundles.The only two vector bundles with base space B a circle and 1D fibreF are the Mőbius band and the annulus, but the classification of all thedifferent vector bundles over a given base space with fibre of a given dimensionis quite difficult in general. For example, when the base space isa high–dimensional sphere and the dimension of the fibre is at least three,then the classification is of the same order of difficulty as the fundamentalbut still largely unsolved problem of computing the homotopy groups ofspheres [Hatcher (2002)].Now, there is a natural direct sum operation for vector bundles overa fixed base space X, which in each fibre reduces just to direct sum ofvector spaces. Using this, one can get a weaker notion of isomorphism ofvector bundles by defining two vector bundles over the same base space Xto be stably isomorphic if they become isomorphic after direct sum withproduct vector bundles X ×R n for some n, perhaps different n’s for the twogiven vector bundles. Then it turns out that the set of stable isomorphismclasses of vector bundles over X forms an Abelian group under the directsum operation, at least if X is compact Hausdorff. The traditional notationfor this group is ˜KO(X). It is the basis for K–theory (see below). In thecase of spheres the groups ˜KO(S n ) have the quite unexpected propertyof being periodic in n. This is called Bott Periodicity, and the values of˜KO(S n ) are given by the following table [Hatcher (2002)]:n mod 8 1 2 3 4 5 6 7 8˜KO(S n ) Z 2 Z 2 0 Z 0 0 0 ZFor example, ˜KO(S 1 ) is Z 2 , a cyclic group of order two, and a generatorfor this group is the Mőbius bundle. This has order two since the directsum of two copies of the Mőbius bundle is the product S 1 × R 1 , as one cansee by embedding two Mőbius bands in a solid torus so that they intersect