By Morgan J.W., Lamberson P.J.

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6. (Mayer-Vietoris for Singular Cohomology) Suppose X = U ∪ V with U, V open. Then we have the following long exact sequence, j ∗ +j ∗ i∗ −i∗ V V U U −−→ H n (U ∩ V ) −−−−→ · · · −−→ H N (U ) ⊕ H N (V ) −− · · · −−−−→ H n+1 (U ∩ V ) −−−−→ H n (X) −− where jU : U → X, jV : V → X, iU : U ∩ V → U and iV : U ∩ V → V are the inclusions. Proof. We have the short exact sequence of chain complexes, (jU )∗ +(jV )∗ (iU )∗ −(iV )∗ 0 −−−−→ S∗ (U ∩ V ) −−−−−−−−→ S∗ (U ) ⊕ S∗ (V ) −−−−−−−−→ S∗small (X) −−−−→ 0 Dualizing, we obtain, 48 (iU )∗ +−(iV )∗ (jU )∗ +(jV )∗ ∗ (X) −−−−−−−−→ S ∗ (U ) ⊕ S ∗ (V ) −−−−−−−−−→ S ∗ (U ∩ V ) −−−−→ 0 0 −−−−→ Ssmall This gives rise to a long exact sequence in cohomology.

We proceed by induction on n. Suppose that we have Hk (S n−1 ) = Z k = n − 1, 0 0 otherwise for some n − 1 ≥ 1. Choose a point p ∈ S n and let p∗ be the antipodal point. Let U = S n − {p} and V = S n − {p∗ }. Then {U, V } is an open cover of S n . Applying Mayer-Vietoris, we obtain the long exact sequence: · · · → Hk (U ∩ V ) → Hk (U ) ⊕ Hk (V ) → Hk (S n ) → Hk−1 (U ∩ V ) → · · · . Both U and V are homeomorphic to Rn and hence are contractible. Then by the homotoppy axiom, Z ∗=0 H∗ (U ) = H∗ (V ) ∼ = 0 otherwise As an exercise, show that, U ∩ V = S n − {p} − {p∗ } is homotopy equivalent to S n−1 , and so by the homotopy axiom and the inductive hypothesis, H∗ (U ∩ V ) ∼ = = H∗ (S n−1 ) ∼ Z ∗ = 0, n − 1 0 otherwise Putting these into the Mayer-Vietoris long exact sequence, we see that for k ≥ 2.

Let (X, A) be a pair of topological spaces. Dual to the short exact sequence 0 → S∗ (A) → S∗ (X) → S∗ (X, A) → 0 is the short exact sequence 0 → S ∗ (X, A) → S ∗ (X) → S ∗ (A) → 0. Where S ∗ (X, A) is defined to be the kernel of the map induced by the inclusion, i∗ : S ∗ (X) → S ∗ (A). This dual sequence is exact since S∗ (X, A) is free abelian, and hence the first short exact sequence splits. 5. (The Long Exact Sequence of a Pair for Singular Cohomology) For a pair of topological spaces (X, A), there is a long exact sequence in cohomology: β · · · −−−−→ H k (X, A) −−−−→ H k (X) −−−−→ H k (A) −−−−→ H k+1 (X, A) −−−−→ · · · where the first three maps are induced by the inclusions and β is the connecting homomorphism associated to the above short exact sequence of chain complexes.

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