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It is shown that a map/: X — F(A, W) is a cofibration if its adjoint/: X A A -» W is a cofibration and X and A are locally equiconnected (LEC) based spaces with A compact and nontrivial. Thus, the suspension map r¡: X -» Ü1X is a cofibration if X is LEC. Also included is a new, simpler proof that C.W. complexes are LEC. Equivariant generalizations of these results are described. In answer to our title question, asked many years ago by John Moore, we show that 7j: X -> Í22A is a cofibration if A is locally equiconnected (LEC)—that is, the inclusion of the diagonal in A X X is a cofibration [2,3]. An equivariant extension of this result, applicable to actions by any compact Lie group and suspensions by an arbitrary finite-dimensional representation, is also given. Both of these results have important implications for stable homotopy theory where colimits over sequences of maps derived from r¡ appear unbiquitously (e.g., [1]). The force of our solution comes from the Dyer-Eilenberg adjunction theorem for LEC spaces [3] which implies that C.W. complexes are LEC. Via Corollary 2.4(b) below, this adjunction theorem also has some implications (exploited in [1]) for the geometry of the total spaces of the universal spherical fibrations of May [6]. We give a simpler, more conceptual proof of the Dyer-Eilenberg result which is equally applicable in the equivariant context and therefore gives force to our equivariant cofibration condition. The key to the simplified proof is Proposition 2.5, a result of independent interest on cofibrations joining a pair of pushout diagrams. Our definitions and conventions on spaces and cofibrations are contained in §1. §2 contains the statements of our results and §3 is devoted to their proofs. §4 sketches the equivariant extensions of our theorems. 1. Preliminaries on spaces and cofibrations. We work in the category of compactly generated weak Hausdorff spaces (see [4,7,8,11,13]). Products, smash products, function spaces (written exponentially), and based function spaces (written F(A, A)) are given the topologies appropriate to this category. A map /: A -» Y is called a Received by the editors January 29, 1980 and, in revised form, June 24, 1981. 1980 Mathematics Subject Classification. Primary 55P05; Secondary 55Q10, 55R15, 55R35, 55U35, 55U40. Kev words and phrases. Cofibration, equivariant cofibration, suspension map, locally equiconnected space, LEC space, Lillig Union Theorem, Dyer-Eilenberg Adjunction Theorem, geometric realization. '1982 American Mathematical Society (X)O2-9947/82/OO0O-OO22/$O4.0O Í22A

When is the natural map 𝑋→ΩΣ𝑋 a cofibration?