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X with f ∈ M, there is a unique morphism h : B → X with hf = g. The orthogonal subcategory problem [13] asks whether D is reflective in C, i.e., under which conditions the inclusion functor D → C admits a left adjoint E : C → D; see [17]. Many authors have given conditions on the category C and the class of morphismsM ensuring the reflectivity of D, sometimes even providing an explicit construction of the left adjoint E : C → D; see for example Adams[1], Bousfield [3],[4], Deleanu-Frei-Hilton [9][10], Heller [15], Yosimura [22], Dror-Farjoun [11], Kelly [12]. The functor E is often referred to as a localisation functor of C at the subcategory D. Most of the known existence results of left adjoints work well when the category C is cocomplete [12] or complete [19]. Unfortunately, these methods cannot be directly applied to the homotopy category of CW-complexes, as it

A further advance—though it was not presented as such—was contained in Theorem 1.6 of [1], which effectively gave an example showing that the answer to our question is not always affirmative. The main purpose of this paper is to generalize both Theorem 0.1 and the counterexample contained in Theorem 1.6 of [1]. Thus we prove (cf. Theorems 2.1 and 2.3, which reproduce part (b) and part (a) of Theorem 0.2 respectively)