NOTE ON FIBRE SPACES

Abstract

This chapter presents an assumption where W2n–1 is the space of unit tangent vectors to the n-sphere Sn. Then, W2n–1 is a fiber bundle over Sn, with fiber Sn–1. When n is odd, W2n–1 has a cross-section and its homotopy type is indistinguishable by any of the known algebraic invariants from that of Sn ×Sn–1. J.P. Serre proposed the problem of determining whether or not W9 is of the same homotopy type as S5 × S4. The chapter presents the solution of the problem by the following theorem, that states that W2n–1 is of the same homotopy type as Sn × Sn–1 if and only if π2n+1 (Sn+1) has an element with Hopf invariant unity. It is also proved in the chapter that a fiber space of which a fiber is a retract is of the same homotopy type as the topological product of the base space and the fiber.