Collapsing Theory of 4-manifolds and the geometry of Alexandrov Spaces

点击翻译按钮获取中文摘要

In this research from 1998 to 2000, we have decided the topology of collpsed Riemannian closed 4-manifolds with uniform lower bound on sectional curvature and upper diameter bound in terms of the singular fiber structure over the limit Alexandrov space. In the course of the proof of this result, we have also succeeded the classification of complete nonnegatively curved Alexandrov spaces of ditmensions three and four. The detail of the results are the following :・We have constructed a local S^1-action on a collapsed 4-manifold when the limit space is of dimension three ;・We have proved that a collapsed 4-manifold is either a sphere-bundle or a Seifert torus-bundle over the limit space ;・In the case when the limit has dimension two and non-empty boundary, a decomposition of the collapsed 4-manifold along the boundary of the limit space has been obtained ;・In the case when the limit is an interval, we have proved that the collapsed 4-manifold is a gluing of at least two and at most four disk-buundles ;・We have proved a soul theorem for 4-dimensional concompact complete Alexandrov spaces with nonnegative curvature ;・We have succeeded in proving a splitting theorem for complete Alexandrov spaces with nonnegative curvature and with disconnected boudary ;・We have obtained a metric classification of compact Alexanadrov spaces with nonnegative curvature and with maximal vertices.