Collapsing 4-manifolds under a lower curvature bound

Abstract

In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions. This enables us to understand the set of homeomorphism classes of closed orientable 4-manifolds with those geometric bounds on curvature and diameter. In the course of the proof of the above results, we obtain the soul theorem for 4-dimensional complete noncompact Alexandrov spaces with nonnegative curvature. A metric classification for 3-dimensional complete Alexandrov spaces with nonnegative curvature is also given.

Theorems & Definitions (302)

• Theorem 1
• Theorem 2
• Remark 3
• Remark 4
• Corollary 5
• Theorem 6
• Theorem 7
• Theorem 8: FY:fundgp
• Theorem 9
• Corollary 10