Induced fibrations and cofibrations

Abstract

Introduction. It is well known that any map is homotopically equivalent to a fiber map, i.e., to the projection of the total space on the base in a fibration. Simple examples, however, reveal that there are maps which fail to be homotopically equivalent to any inclusion of a fiber in the total space, and the problem of characterizing the maps which are equivalent to such inclusions was first raised in [14]. By the first remark, the map under consideration may be assumed from the beginning to be a fiber map p: E -B, and the problem is thus converted into that of characterizing the fibrations p which are equivalent to induced fibrations. There are two immediate necessary conditions: the fiber F of p must have the homotopy type of some loop space fl Y, and the inclusion i: F -* E must map the generalized homotopy group 7r(X, fQF) into the center of n(X, QE) for any space X; the latter is a mild generalization of the well known fact that the boundary operator in the homotopy sequence of a fibration E -* B -* Y maps XT2( Y) into the center of 7r1(E). The first result in this direction is due to Serre and asserts that p is an induced fibration if B is 1-connected and F has a single nonvanishing (Abelian) homotopy group. This result was generalized [6], [12], [17], [15] to allow F to have at most m -1 nonvanishing homotopy groups in consecutive dimensions provided it has the homotopy type of a loop space and B is (m -)-connected. In the first two sections of this paper we give results which allow F to have 2m -1 nonvanishing homotopy groups in consecutive dimensions provided B is (m 1)-connected and F has the homotopy type of a loop space under a homotopy equivalence fulfilling a certain condition which involves the operation of QB on F. In case F has m nonvanishing homotopy groups in consecutive dimensions, the sufficient condition in order that p be induced is expressed by means of the vanishing of a certain Whitehead product; this result answers a question raised in [15]. Dually, any map is homotopically equivalent to an inclusion A -> X having the homotopy extension property, but few maps are equivalent to identification maps resulting by shrinking to a point such a subset A of X. This leads to the problem of characterizing the cofibrations A -> X which are induced by maps of some space Y into A. Our results extend the range of applicability of some previous results [8], [12] by imposing an additional condition which involves the cooperation of the suspension EA on the cofiber C obtained from X by shrinking A to a point; the condition is conveniently expressed in terms of the Hopf invariant of a cofibration