COMPLETE RIEMANNIAN MANIFOLDS 9 It is easy to check that bornotopy is an equivalence relation. The terminology is suggested by homotopy theory the link may become clearer if one observes that the definition of bornotopy may be reformulated as follows: / and g are bornotopic if there is a morphism h:Mx{0,l}—•i V with / = /ioz0, j = fto^, where io,i\: M —• M x {0,1} are the obvious inclusions. Of course, since we are working with Borel functions, the topological structure is irrelevant we can replace {0,1} by [0,1] (or by any compact metric space with two distinguished points) without changing the definition. Two spaces which are eventually Lipschitz equivalent in the sense of [27] are bornotopy-equi valent. An example of bornotopy-equivalence that will be useful later is the following. Let N be a, space and M a (closed) subspace. Recall from [54] the notation Pen(M R) = {x G N: d{x,M) R}. We will say that M is co-dense in N if there is R 0 such that N = Pen(M R). (2.6) PROPOSITION: If M is uj-dense in N, then the inclusion map i: M —• N is a bornotopy-equivalence. PROOF: Suppose that Pen(M R) = N\ then for any y € N there is x G M with d(x,y) 2R. Define a function p: N — * M to be a Borel selection of such an a?, in other words p is a Borel map such that d(x,p(x)) 2R\ it is easy to check that such a Borel selection can be made. Then p is a morphism, and p o i ~ 1M, i ° p ~ ITV, SO i is a bornotopy-equivalence. • We will discover in Chapter 3 that coarse cohomology behaves in the same way with respect to bornotopy as ordinary cohomology does with respect to homotopy. 2.2. Definition of coarse theory In the definition of coarse theory, we will use the following notations. Let M be a space. Then Mq+l denotes the Cartesian product of q-\-l copies of M, and A or more fully Aq(M) C Mq+1 denotes the multi-diagonal {(#,... , x) : x G M}. For definiteness, we must specify the metric that we are using on Mq+l: we define it by d((x0i... ,Xfl),(j/0,... ,yq)) = max{d(x0,t/o),... } d(xqiyq)}. A function p: M9+l —• R is said to be locally bounded if it is bounded on every compact subset. The support of p is denoted Supp(y?). (2.7) DEFINITION: Let M be a space. The coarse complex CX*(M) is defined as follows: CXq(M) is the space of locally bounded Borel functions p: Mq+1 — * R which satisfy the following support condition: for each R 0, the set Supp(v?)nPen(A iJ) is relatively compact in Mq+l.

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