Coarse Fiber Structure,Coarse Embedding At Infinity And Higher Index

Coarse geometry is the important research content in noncommntative geometry,and it mainly studies the structure of large scale geometry of metric spaces.The key problem in coarse geometry is the coarse Baum–Connes conjecture.Coarse geometry has many important implications in geometry topology,geometric group theory,such as Novikov conjecture,Borel conjecture,Gromov zero-spectral conjecture and so on.In this thesis,we study coasre fibration of metric spaces,coarse embedding at infinity and higer index.Recently,J.Deng,Q.Wang and G.Yu introduced the “A-by-CE” structure for group extensions,and proved that for a sequence of group extensions with bounded geometry and “A-by-CE” structure,the coarse Baum–Connes conjecture holds for their coarse disjoint.Their result implies that the coarse Baum–Connes conjecture holds for many relative expanders.The authors propose the following question: Does the coarse Baum–Connes conjecture hold for bounded geometry metric space with “CE-by-CE”structure? In this thesis,we partially answer the above question.We prove that the coarse Novikov conjecture holds for metric spaces with “CE-by-CE” structure.In the third chapter,wo introduce the “A-by-CE” coarse fibration,and study higher index problem on bounded geometry metric spaces with “A-by-CE” coarse fibration,which contain abound examples that can’t coarsely embedd into Hilbert space.We prove that the maximal and reduced Roe algebras of bounded geometry metric spaces with “A-by-CE” coarse fibration have the same K-theory.As a consequence,we prove that the(maximal)coarse Baum–Connes conjecture holds for bounded geometry metric spaces with “A-by-CE” coarse fibration.In the last chapter,we introduce the notion of coarse embedding into Hilbert space at infinity,and then we introduce the obstruction to coarse embedding at infinity.Finally,we prove that for a sequence of finite graphs with bounded geometry,if there exists an asymptical faithful Galois covers,which admits a coarse embedding into Hilbert space,then the coarse Novikov conjecture holds for the coarse disjont union of the sequence of finite graphs.The result implies that the coarse Novikov conjecture holds for any box spaces of finitely generated,residually finite groups which are coarsely embeddable into Hilbert space,and also holds for a sequence of finite graphs with bounded geometry and large girth.