CHAPTER 2 Basic properties of coarse cohomology 2.1. Th e uniformly homologous categories The coarse cohomology groups HX*{M) will be defined for metric spaces M. However, they will not be functorial under all continuous maps of such spaces, and they will be functorial under certain non-continuous maps. In this section we will define the appropriate categories on which HX* will be functorial. (2.1) DEFINITION: Let M and N be metric spaces, and let f': M —• N be a function, not necessarily continuous. The function f is called uniformly homol- ogous if, for every R 0, there exists S 0 such that Vx,x' e M,d(x,x') R= d(f(x)J(x')) S. REMARKS: (i) Here, and throughout this paper, we use the letter d to denote the dis- tance in a metric space. Also, the notation B(x e) will denote the open ball of centre x and radius e. (ii) What we have done is to write down the definition of uniform continuity with the usual e and 6 the wrong way round. Of course one should think of R and S as being large rather than small the content of the definition is that one has uniform control on the expansiveness of the function / . (iii) The function / : Z + Z defined by f(n) n2 is an example of a uni- formly continuous function that is not uniformly homologous. (iv) A function which is eventually Lips chit z in the sense of Ferry and Peder- sen [27] is uniformly homologous for general spaces the converse need not be true, but most of the uniformly homologous functions we are interested in will be eventually Lipschitz. (v) It is straightforward to show that the composite of two uniformly homol- ogous functions is again uniformly homologous. (vi) We define a borneomorphism as a uniformly homologous bijection whose inverse is also uniformly homologous. (vii) The 'borno' terminology is meant to suggest that we are studying bound- ed sets instead of the open sets of ordinary topology. 7
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