K-Homology At Infinity And Higher Index Problems On Metric Spaces

Coarse geometry and(higher)index theory are key topics in non-commutative geometry,the core problem in coarse geometry is the coarse Baum–Connes conjecture,which has many significant applications in geometry,topology,and geometric group theory.In this thesis,we study the coarse fibration structure on metric spaces and the coarse Novikov conjecture of certain spaces,introduce the notion of K-homology at infinity,and study the higher index problem at infinity.In the first part,we introduce the“CE-by-(H)”coarse fibration structure and the“CEH-by-CEM”group extension structure,respectively,and prove the coarse Novikov conjecture for the discrete space with bounded geometry which admit the“CE-by-(H)”coarse fibration structure and the countable discrete group satisfying the“CEH-by-CEM”group extension structure.In the second part,we introduce the concept of K-homology at infinity,and estab-lish the index map at infinity,transforming the higher index problem of space into the higher index problem at infinity.As an application,we prove that the coarse Novikov conjecture holds for discrete space with bounded geometry that can be fibred coarsely embedded into an ~pspace.This applies to all box spaces of a residually finite hyper-bolic group and a large class of warped cones of a compact space with an action by a hyperbolic group.