The Homotopy Theory of Sphere Bundles Over Spheres (I)

Abstract

Publisher Summary This chapter discusses the homotopy theory of q -sphere bundles over n -spheres ( n , q ≥ 1). When q = 1 or n = 1 the equivalence class, in the sense of fiber bundle theory, of such a bundle is characterized by its homology groups, which are homotopy invariants. Therefore, the classification of such bundles into homotopy types is the same as their classification by bundle equivalence. When q ≥ 1, there are precisely two types of q -sphere bundles over S 1 , namely, the orientable product bundles and the non-orientable generalized Klein bottles. All q -sphere bundles over S 1 ( q ≥ 1) admit cross-sections. When n > 2, any 1-sphere bundle over S n is a product bundle and the 1-sphere bundles over S 2 are the lens spaces of type ( m , 1).