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.