Total curvatures of model surfaces control topology of complete open manifolds with radial curvature bounded below. II

Abstract

We prove, as our main theorem, the finiteness of topological type of a complete open Riemannian manifold $M$ with a base point $p\in M$ whose radial curvature at $p$ is bounded from below by that of a non-compact model surface of revolution $\tilde{M}$ 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 $\tilde{p}\in \tilde{M}$. Notice that our model $\tilde{M}$ 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.

arXiv comment37 pages, No figures, Submitted to Transactions of Amer. Math. Soc. (December 17, 2007), The [v1] is the 2nd revised version (October 17, 2008) according to the 2nd referee's report