Dihedral and quaternionic homology and mapping spaces

We associate to the singular chains of the Moore loop space Y two dihedral chain complexes. The dihedral homology of these complexes is shown to be isomorphic to the O(2)-equivariant homology of the mapping spaces Y ^o(2) and . We also construct simple homotopy theoretic models of EO(2) x _O2 (Y)^G/BO(2) for G = O(2) or S¹, where denotes suspension. Quaternionic analogues of these results are also established.