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

TL;DR

This paper proves that complete open Riemannian manifolds with radial curvature bounded below by a model surface of revolution with finite total curvature have finite topological type, using a new Toponogov comparison theorem, and applies this to Milnor's conjecture.

Contribution

It establishes a finiteness result for the topological type of manifolds under weaker curvature conditions and introduces a novel comparison theorem.

Findings

Finiteness of topological type under radial curvature bounds.

New Toponogov comparison theorem without diameter growth condition.

Partial progress on Milnor's open conjecture.

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.