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 n−p−1 c (X ∩ 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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