0.3. The ultimate duality theorem 7
For a compact pair (A, B), there is a fundamental relationship between
the cohomology with compact supports of A B and the unrestricted co-
homology of that pair (Spanier, 1966, p. 321):
Theorem 0.3.3. For any compact Hausdorff pair (A, B),
ˇ
H
q(A
c
B; G)
∼
=
ˇ
H
q(A,
B; G).
Corollary 0.3.4. A compact, connected, n-dimensional ∂-manifold M is
G-orientable if and only if
Hn(M,
∂M; G)
∼
=
G.
Corollary 0.3.5. If A is a locally compact Hausdorff space and B is a closed
subset of A, then
ˇ
H
q(A
c
B; G)
∼
=
ˇ
H
q(A,
c
B; G).
Proof. Let
A+,B+
denote the one-point compactifications of A, B, respec-
tively, with ∞ the ideal point. Then
ˇ
H
q(A;
c
G)
∼
=
ˇ
H
q(A+,
{∞}; G)
∼
=
ˇ
H
q(A+;
G)
and similarly for B. The Five Lemma and another application of Theorem
0.3.3 yield
ˇ
H
q(A,
c
B; G)
∼
=
ˇ
H
q(A+,B+;
G)
∼
=
ˇ
H
q(A
c
B : G).
Corollary 0.3.6. Every open subset of a G-orientable manifold is G-orient-
able.
Proof. If U is a connected open subset of the G-orientable manifold M
n,
then G
∼
=
H0(U; G)
∼
=
ˇ
H
n(M,
c
M U; G) by duality, and
ˇ
H
n(M,
c
M U; G)
∼
=
Hc
n(U;
G) by 0.3.5.
Corollary 0.3.7. If S is a locally compact Hausdorff space and q ≥ 2, then
ˇ
H
q(S
c
× R; G)
∼
=
ˇ
H
q−1(S;
c
G).
Proof. Let
S+
denote the one-point compactification of S, with ∞ the ideal
point. A Mayer-Vietoris argument shows that
ˇ
H
q(Suspension
of
S+;
G)
∼
=
ˇ
H
q−1(S+;
G) for q ≥ 2. Thus,
ˇ
H
q(S
c
× R; G)
∼
=
ˇ
H
q(Suspension
of
S+,
Suspension of ∞; G)
∼
=
ˇ
H
q(Suspension
of
S+;
G)
∼
=
Hq−1(S+;
G)
∼
=
ˇ
H
q−1(S+,
{∞}; G)
∼
=
ˇ
H
q−1(S;
c
G).
Corollary 0.3.8. A manifold M is G-orientable if and only if M × R is
G-orientable.
