Ja n 20 09 Total Curvatures of Model Surfaces Control Topology of Complete Open Manifolds with Radial Curvature Bounded Below

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold M with a base point p ∈ M whose radial curvature at p is bounded from below by that of a non-compact model surface of revolution M̃ which admits a finite total curvature and has no pair of cut points in a sector. Here a sector is, by definition, a domain cut off by two meridians emanating from the base point p̃ ∈ M̃ . Notice that our model M̃ does not always satisfy the diameter growth condition introduced by Abresch and Gromoll. In order to prove the main theorem, we need a new type of the Toponogov comparison theorem. As an application of the main theorem, we present a partial answer to Milnor’s open conjecture on the fundamental group of complete open manifolds.