8 0. Prequel

The equivalence of Alexander-Spanier cohomology with

ˇ

Cech cohomol-

ogy on compact spaces (Spanier, 1966, p. 319) and computation with the

latter yields:

Theorem 0.3.9 (Dimension). Hc(S)

i

∼

= 0 whenever i dim S and S is

locally compact.

Corollary 0.3.10 (Local Duality). Let M be a G-orientable n-manifold,

X a closed, k-dimensional subset of M, and x ∈ X. If V is a coordinate

neighborhood of x, then Hp(V X; G) = 0 for 0 ≤ p ≤ n − k − 2.

Proof. Let V be a coordinate neighborhood of x. Then

ˇ

H

n−p−1(X

c

∩ V ; G)

∼

=

Hp+1(V, V X; G)

∼

=

Hp(V X; G),

the latter isomorphism coming from the long exact homology sequence of

the pair (V, V X). All groups in this line are trivial, since n − p − 1 k

and k = dim(X) ≥ dim(X ∩ V ). Thus Hp(V X; G)

∼

= 0.

One of the delicate issues that will require attention later in the book

is the question of when the homology connectivity of Corollary 0.3.10 can

be promoted to connectivity in the sense of homotopy. Dimension one is

special in that regard.

Corollary 0.3.11. If X is a k-dimensional closed subset of a connected

n-manifold M, k ≤ n − 2, then M X is pathwise connected. Indeed,

each x ∈ X has arbitrarily small neighborhoods U such that U X is path

connected.

Exercises

0.3.1. If Σ ⊂

Sn

and Σ is homeomorphic to

Sk,

k n, then

Hi(Sn

Σ; Z)

∼

=

Z if i = n − k − 1,

0 otherwise.

0.3.2. (Jordan Separation Theorem) If Σ ⊂

Sn

and Σ is homeomorphic

to

Sn−1,

then

Sn

Σ has exactly two components and Σ is the

topological frontier of each.

0.3.3. If X is a k-dimensional closed subset of a ∂-manifold M

n,

k ≤ n−2,

then M X is connected.

0.3.4. No manifold M

n

that contains a copy of

MB×In−2

(MB = M¨obius

band) is Z-orientable.