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.