Note on Fibre Spaces

Abstract

This chapter presents an assumption where W 2n–1 is the space of unit tangent vectors to the n -sphere S n . Then, W 2n–1 is a fiber bundle over S n , with fiber S n–1 . When n is odd, W 2n–1 has a cross-section and its homotopy type is indistinguishable by any of the known algebraic invariants from that of S n × S n–1 . J.P. Serre proposed the problem of determining whether or not W 9 is of the same homotopy type as S 5 × S 4 . The chapter presents the solution of the problem by the following theorem, that states that W 2n–1 is of the same homotopy type as S n × S n–1 if and only if π 2n+1 ( S n+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.