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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