COMPLETE RIEMANNIAN MANIFOLDS

11

H*(M) as the Alexander-Spanier cohomology with compact supports of M. A

g-cochain in this theory is just an equivalence class of functions z: Mq+l —* R

that are locally zero on the complement of a compact set, two such functions

being considered equivalent if they agree on a neighbourhood of the diagonal (see

Spanier [64, pages 306f and 319f]). We define the map c:

HXq(M)

— H$(M)

by sending a cocycle p to its truncation to any penumbra Pen(A;i2) of the

diagonal.

If M is a manifold and p is a smooth cocycle, then we may also interpret

Hq(M) as de Rham theory. In this interpretation, the map c is induced by

fo®--®fq*-*fodfxA--Adfq

for smooth functions /,-.

(2.12)

LEMMA:

Let i: H*(M) —• H*(M) denote the natural map. Then

LOC=0

as a map HXq{M) -+ Hq(M) for q\.

PROOF:

Let CA*(M) denote the Alexander-Spanier complex for M, and

CA*C(M) the subcomplex defining the cohomology with compact supports. Re-

call (2.10) that C*(M) denotes the acyclic complex of M. There is a commutative

diagram of complexes

CX*{M) - ^ CA*(M)

C*(M) —• CA*(M)

The result now follows from the acyclicity of C*(M). D

Later we will see examples which show that the sequence of maps

0 - HX*(M) A H*(M) -^ H*(M)

need not be exact at either of its middle terms.

(2.13) Functoriality: Coarse cohomology is contravariantly functorial on

the category UBB. In fact, if / : M —* N is a morphism (in the category UBB,

as usual), then it induces a chain transformation /*: CX*(N) —• CX*(M) by

the usual formula

(f*(p)(x0,...

}

xq) = p(f(x0),... J(xq)).

Since / maps bounded sets to bounded sets, f*ip is locally bounded; and it is

clear that /* commutes with d. So all we have to check is that f*p does indeed

satisfy the support condition. Now, let g: Mq+1 —• A^^+1 be the map induced by

/ ; g is proper, so to check that Supp(/*^) D Pen(AA/; R) is relatively compact,

it is enough to check that its image under g, namely Supp(£) fl g(Pen(AM] R)),

is compact. But since g is uniformly homologous, there is S 0 such that

g(Pen(A\f]R)) C Pen(Aw, S)t so this is immediate.