THE HOMOTOPY THEORY OF SPHERE BUNDLES OVER SPHERES (I)

Abstract

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 S1, namely, the orientable product bundles and the non-orientable generalized Klein bottles. All q-sphere bundles over S1 (q ≥ 1) admit cross-sections. When n > 2, any 1-sphere bundle over Sn is a product bundle and the 1-sphere bundles over S2 are the lens spaces of type (m, 1).