We have defined and studied some pseudogroups of local diffeomorphisms which generalise the complex analytic pseudogroups. A 4-dimensional (or 8-dimensional) manifold modelled on these a€?Further pseudogroupsa€? turns out to be a quaternionic (respectively octonionic) manifold.We characterise compact Further manifolds as being products of compact Riemann surfaces with appropriate dimensional spheres. It then transpires that a connected compact quaternionic (H) (respectively O) manifold X, minus a finite number of circles (its a€?real seta€?), is the orientation double covering of the product Y ?— P2, (respectively Y?—P6), where Y is a connected surface equipped with a canonical conformal structure and Pn is n-dimensonal real projective space.A corollary is that the only simply-connected compact manifolds which can allow H (respectively O) structure are S4 and S2 ?— S2 (respectively S8 and S2?—S6).Previous authors, for example Marchiafava and Salamon, have studied very closely-related classes of manifolds by differential geometric methods. Our techniques in this paper are function theoretic and topological.
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