8 0. Prequel
The equivalence of Alexander-Spanier cohomology with
ogy on compact spaces (Spanier, 1966, p. 319) and computation with the
Theorem 0.3.9 (Dimension). Hc(S)
= 0 whenever i dim S and S is
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
∩ 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)
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
0.3.1. If Σ ⊂
and Σ is homeomorphic to
k n, then
Z if i = n − k − 1,
0.3.2. (Jordan Separation Theorem) If Σ ⊂
and Σ is homeomorphic
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
k ≤ n−2,
then M X is connected.
0.3.4. No manifold M
that contains a copy of
(MB = M¨obius
band) is Z-orientable.