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In this note we give definitions of twisted generalized homology and cohomology, in which the twisting is defined by spherical fibrations, and we prove that under certain conditions Poincaré duality holds. The definitions are chosen such that when the fibrations are trivial, Whitehead's definitions, as given in [6], hold good. We use the notation of [ó] and in the future refer to twisted generalized homology and cohomology simply as homology and cohomology and to spherical fibrations as fibrations. Full details of the work reported here will appear elsewhere. I should like to thank Professor C. T. C. Wall for suggesting this work and for his guidance and criticism.

Duality in a twisted homology theory