Full title: "Geometry, conformal structure and lassification of CMC surfaces via real analytic topology" ABSTRACT: A real analytic variety V carries a (mod 2) fundamental class, so any proper map from V to a connected manifold of the same dimension has a well-defined (mod 2) degree. We prove existence, uniqueness, duality and classification results for constant mean curvature (CMC) surfaces by applying this idea to various proper maps from CMC moduli space to natural classifying spaces. For example, the forgetful map to Riemann moduli space (which forgets all but the underlying punctured conformal structure of the surface) can be supplemented to give such a proper map; we will discuss the extent to which the forgetful map is surjective.