Filtered Ends of Pairs of Groups

Proper and CW-proper homotopy theory as described in Chap. 10 can be regarded as the homotopy theory of maps which preserve a finite (or finite type) filtration. In this chapter we introduce a generalization in which the filtration is by complexes which are not necessarily of finite type. Although all the main ideas have already been seen in our discussion of proper homotopy, there is need for an exposition of the foundations of the generalized theory. The corresponding homology and cohomology theories of filtered ends are discussed. The immediate occasion is our discussion of an alternative way of counting ends of pairs of groups. That appears in Sect. 14.5.