Extensions of Maps as Fibrations and Cofibrations

Suppose /: X —• Y is a map of 1-connected spaces. In the "stable" range, roughly where the connectivity of Y exceeds the homology, or homotopy, dimension of X, it is well known that / can be extended as a cofibration C — X — Y, or respectively a fibration X — Y — B. A criterion is given for the existence of such extensions in a less restrictive "metastable" range. A main result is that if / is at least 2-connected and 2 con Y > dim Y 1, dim X, then / extends as a cofibration if and only if the map (1 X /)A : X — (X x Y)/X factors through /. We consider the question: Given a map /: X —• Y, when can it be extended up to homotopy to a fibration X —• Y —♦ B, or a cofibration C —• X — Y? Generally no such extension is possible. In an appropriate "stable" range of dimensions and connectivities the extension can be made. The object of this paper is to give a necessary and sufficient condition for the extension in the "metastable" range. Only the simply connected version is considered here, and spaces are understood to have the homotopy type of a CW complex; basepoints are nondegenerate. A cofibration lemma similar to 1.2 was announced in [3], and the nonsimply connected version is given in [4]. It was developed for use as a main step in constructing a surgery theory for Poincaré' spaces. Corollary 1.3 and some standard Spanier-Whitehead duality can be used to do surgery on simply connected Poincaré' spaces. T. Ganea [2] and R. Nowlan [6] have similar results, but their extension criterion involves operation, or "cooperation" of an ¿-space, or co-A-space. Our criterion is based, in the cofibration case, on a homotopy analog of the vanishing of certain products in cohomology. See the comments after 1.3. 1. Statements of results. First some notations are established. For a map /: X — y we denote by Y. the range of / converted functorially into Received by the editors February 18, 1974 and, in revised form, August 26, 1974. AMS (MOS) subject classifications (1970). Primary 55D05.