On Fibre Spaces and the Evaluation Map

Abstract

In [6], the author studied a subgroup G(X) of the fundamental group w1(X). This group may be defined by means of the evaluation map co: L(A, B) B where L(A, B) is the space of continuous maps from A to B. The definition of G(X), called the evaluation subgroup of X, along with some of its properties, appears in ? 2 for the convenience of the reader. The main theme of this paper is the investigation of the role played by the evaluation subgroup in the theory of Hurewicz fibrations. In ? 3, given a Hurewicz fibration p: Em B, with B a cw-complex and with fibre F, we classify the set of fibre homotopy equivalences from E to E up to homotopies of fibre homotopy equivalences. In fact, this set forms a group, 2(E), with multiplication induced by composition. Then if Bo. is the classifying space for Hurewicz fibrations with fibre F, and k: B Bo. is the classifying map for p: E m B (see [1] or [3]), we have Theorem 1, which says that 2(E) w1(L(B, Boo); k). Fire use the results of ? 3 to study the evaluation subgroup of the fibre F in ? 4. With every fibration p: E m B there is a homomorphism d: wr2(B) ) w1(F) which arises from the homotopy exact sequence for p. The union, over all fibrations with fibre F, of the images of d will be shown to equal G(F). As a consequence, given a Hurewicz fibration p: E B, every map f: S2 B can be lifted to f: S2 E if the fibre F is a compact, connected, polyhedron with Euler-Poincar6 number different from zero. In fact, even a much more general statement is true. In ? 5, we take a very different tack. For every fibration p: E B with fibre F, we may ask the following question. Which homotopy equivalences of F into itself can be extended to fibre homotopy equivalences of E into itself? That is, which homotopy equivalences can be extended to fibre preserving maps from E to E which send each fibre of E into itself? The homotopy classes of these homotopy equivalences form a subgroup of the group of homotopy equivalences from F to F. We denote this group by WY(E). Using the result of ? 3, we show (Theorem 3) that 7(E.) is isomorphic to G(B.). Here p,: EB. Bo. is a universal fibration for the fibre F. Using the results of ? 2, we show there exists a connected F such that