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.
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