The Geometry And Topology Of Complete Manifolds With Certain Curvature Bounds

In this paper,I will mainly concern the topology and geometry of complete non-compact manifolds with nonnegative Ricci curvature,including some finiteness result under the condition of the interior diameter growth and asymptotically nonnegative cur-vature and also some examples with infinite topology.Abresch and Gromoll proved that an n dimensional complete noncompact manifold with nonnegative Ricci curva-ture and interior diameter growth o(r1/n)must be of finite topological type provided with sectional curvature bounded below.The author gave a generalization of their re-sult by proving that an n dimensional complete noncompact manifold with nonnegative Ricci curvature and 2?th asymptotically nonnegative curvature(0???1)must be of finite topological type provided with interior diameter growth o(r(n-1)?+1/n).On the other hand,the examples with infinite topology constructed by the author consist of two family of manifolds,which both admit positive Ricci curvature and quadratically asymptotically nonnegative curvature,however the ideas of their construction are dif-ferent.One of the examples is an adjustment of Menguy's examples[1]with infinite topology,whose dimension must be bigger than or equal to 6.The other one adopts the topological construction of J.P.Sha and D.G.Yang's example with infinite topology but constructs the metric in a totally different way,whose dimension must be bigger than or equal to 5.Both of the examples will apply a gluing criteria came up with by Perelman when constructing a family of compact manifolds with positive Ricci curva-ture.They answer a question raised by J.P.Sha and Z.M.Shen in dimension bigger than or equal to 5 for the first time,namely when n?5,there exists an n dimension-al complete Riemannian manifold with nonnegative Ricci curvature and quadratically asymptotically nonnegative curvature,which is of infinite topological type.In partic-ular,the construction of the second example tells us:for all I?5 and 2?j?I-2,there exists an I dimensional complete Riemannian manifold with nonnegative Ricci curvature,quadratically asymptotically nonnegative curvature and infinite Betti num-ber bj.