Soul Theorem for 4-dimensional Topologically Regular Open Nonnegatively Curved Alexandrov Spaces

TL;DR

This paper proves a Soul Theorem for 4-dimensional topologically regular open Alexandrov spaces with nonnegative curvature, advancing understanding of their topology and potential applications in collapsing 4-manifolds.

Contribution

It establishes a Soul Theorem for 4D topologically regular open Alexandrov spaces, a significant extension of classical results to this broader setting.

Findings

Proves the existence of a soul in 4D Alexandrov spaces with nonnegative curvature.

Provides tools using gradient flows and critical point theory for analyzing topology.

Lays groundwork for future studies on collapsing 4-manifolds.

Abstract

In this paper, we study the topology of topologically regular 4-dimensional open non-negatively curved Alexandrov spaces. These spaces occur naturally as the blow-up limits of compact Riemannian manifolds with lower curvature bound. These manifolds have also been studied by Yamaguchi in his preprint [Yam2002]. Our main tools are gradient flows of semi-concave functions and critical point theory for distance functions, which have been used to study the 3-dimensional collapsing theory in the paper [CaoG2010]. The results of this paper will be used in our future studies of collapsing 4-manifolds, which will be discussed elsewhere.