6 0. Prequel

0.3. The ultimate duality theorem

Homology and cohomology are powerful tools in the study of embeddings.

One reason for their usefulness is the many duality theorems they satisfy that

relate the homology/cohomology of an embedded object to the cohomol-

ogy/homology of its complement. The duality theorems found in (Munkres,

1984), which apply to triangulable homology manifolds, are not quite gen-

eral enough for our purposes, so we state the theorems to be employed. The

following basic duality theorem can be found on page 342 of (Spanier, 1966).

Theorem 0.3.1 (Duality). Let G be an abelian group, and let (A, B) be a

pair of closed subsets of the G-orientable n-manifold M. Then for all p ≥ 0

there exists a natural isomorphism

Hn−p(M B, M A; G)

∼

=

− →

ˇ

H

p(A,

c

B; G).

The group on the right is called the pth Alexander (relative) cohomol-

ogy group of (A, B) with G-coeﬃcients and compact supports. (Throughout

this section G will denote an Abelian group.) When A and B are com-

pact,

ˇc

H

p

(A, B; G) equals the more usual pth Alexander cohomology of the

pair (again with G-coeﬃcients), and when, in addition, both A and B are

themselves complexes, manifolds, or absolute neighborhood retracts (to be

defined in §0.6), it equals the usual singular cohomology of the pair. A

connected n-manifold M is G-orientable if Hc

n(M;

G)

∼

= G; an arbitrary

n-manifold is G-orientable if each of its components is. Every n-manifold is

Z2-orientable; those that contain a copy of MB × In−2, where MB denotes

the M¨ obius Band, fail to be Z-orientable.

Say that a ∂-manifold W is G-orientable if Int W is.

Corollary 0.3.2 (Poincar´ e-Lefschetz Duality). If W is a G-orientable n-

dimensional ∂-manifold, then Hn−p(W ; G)

∼

= Hc

p

(W, ∂W ; G) for every p.

Proof. It follows from Collaring Theorem 2.4.10, to be proved later, that

the manifold W = W ∪ ∂W × (−1, 0], obtained from the disjoint union of

W and ∂W × (−1, 0] by attaching the product ∂W × (−1, 0] to ∂W ⊂ W

along ∂W × {0} in the obvious way, is homeomorphic to Int W and, thus, is

G-orientable. Apply the duality theorem in W with A = W and B = ∂W

to obtain

Hc

p(W,

∂W ; G)

∼

= Hn−p(W ∂W, W W ; G)

= Hn−p(Int W ∪ ∂W × (−1, 0),∂W × (−1, 0); G)

∼

= Hn−p(Int W ; G)

∼

= Hn−p(W ; G).