Quantitative K-theory For Coarse Spaces And Its Applications

One of the most important question in coarse geometry is the coarse Baum-Connes conejecture.And one of the effective way to study coarse Baum-Connes conjecture is quan-titative theory.Quantitative theory is much reflexible than K-theory.We can use quantitative Mayer-Vietories sequence to prove the coarse Baum-Connes conjecture or using propagation in quantitative K-theory to find obstructions for coarse Baum-Connes conjecture.This article is divided into four chapters.In first chapter,we introduce the background and the basics in quantitative K-theory.Mainely about the Bott periodicity and the six term exact sequence.In second chapter,we construct the quantitative maximal coarse Baum-Connes as-sembly map.And we give the relationship between quantitative maximal coarse Baum-Connes conjecture and maximal coarse Baum-Connes conjecture.In third chapter,we introduce persistence approximation property in quantitative K-theory.Persistence approximation property has a very strong relationship with Baum-Connes conjecture with coefficients.We mainly study the relationship between maximal coarse Baum-Connes conjecture with coefficients and persistence approximation property.We prove that that:If a discrete metric X with bounded geometry can fibre coarse embedding into Hilbert space and X is coarsely uniformly contractible,then the maximal Roe algebra for this space has persistence approximation property.One of the most important example which can be fibred coarse embedding into Hilbert space comes from the box space of residually finite groups.The box space of residually finite group can fibred coarse embedding into Hilbert space if and only if the group has Haagerup property[7].By this result we prove that:for a finite generated group,if this group has Haagerup property and admits a cocompact universal example for proper actions,then the maximal Roe algebra of the box space with this residually finite group has persistence approximation property.In fourth chapter,we construct the quantitative maximal coarse Baum-Connes as-sembly map for a family of metric space.Using the quantitative K-theory,we prove that:if the quantitative maximal coarse Baum-Connes conjecture for a family of metric space is true,then the maximal coarse Baum-Connes conjecture for coarse disjoint union of this family of metric spaces holds.