Abstract :

Consider a fibre bundle E over a sphere S^n with fibre F. The n-sphere S^n can be covered by two open sets U and V which contain the two closed hemispheres of the sphere. Over each of the open sets, the bundle is trivial and the total space E of the bundle can be obtained by attaching the trivial bundles U×F and V×F along their intersection which is homeomorphic to the (n-1)-sphere S^{n−1}. For this to work we need to specify how we will identify the fibres over the intersection of the open sets, this is done using a map f : S^{n−1} → G, where G is the group of transformations of the space F. We see that f determines an element in the homotopy group π_{n−1}(G). We will show that the equivalence classes of bundles over S^n (with the same fibre) are in one-one correspondence with the group π_{n−1}(G).