A modified Dold-Lashof construction that does classifyH-principal fibrations

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The classical Dold-Lashof construction [5] and its various modifications ([10] and [19-1), do not in general provide a classifying space B n and a fibration Pn, E n ~ B n that classifies the H-principal fibrations for an H-space H. In [8] we constructed a simple machinery to classify locally homotopy trivial principal fibrations with fiber H, provided a reasonable classifying space is available. In the following pages we propose an alteration of the Dold-Lashof construction to obtain such a classifying space B u and a fibration Pn : E n ~ B u , which is locally homotopy trivial (and therefore allows induced fibrations and classification). E n turns out to be contractible and B n is of the same homotopy type as the Dold-Lashof construction. Thus the relationship to the Milgram-Steenrod construction remains undisturbed. In the case of locally homotopy trivial fibrations whose fiber F is a finite complex we will see that fiber homotopy equivalence is linked in the classical way to principal fiber-homotopy equivalence of the associated H-principal fibration. (Section 7.). In Section 8 we describe a one to one correspondence between the H-principal fibrations and G-principal fiber bundles for certain H-spaces and topological groups G. Maybe we should also point out the relative simplicity of the proof of the classification theorem in section 4, this proof applies of course to principal G-bundles.