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.
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