Abstract
We prove that the space of SU(2) hyperbolic monopoles based at the centre of hyperbolic space is homeomorphic to the space of (unbased) rational maps of the two-sphere. The homeomorphism extends to a map of the natural compactifications of the two spaces. We also show that the scattering methods used in the study of monopoles apply to the configuration space for hyperbolic monopoles giving a homotopy equivalence of this space with the space of continuous self-maps of the two-sphere.
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