Use the preceding exercise to derive relative Mayer–Vietoris sequences for CW pairs \((X, Y) = (A \cup B, C \cup D)\) with \(A = B\) or \(C = D\).
Solution
1. The preceding exercise (E-HAT-2.2-38) gives: from a commutative diagram with exact rows and every third vertical map an isomorphism, one obtains a long exact sequence \[\cdots \to E_{n+1} \to B_n \to C_n \oplus D_n \to E_n \to B_{n-1} \to \cdots.\] Proof: the algebraic lemma.
2. For a CW pair \((X, Y) = (A \cup B, C \cup D)\), the Mayer–Vietoris sequences for the pairs \((A, B)\) and \((C, D)\) fit into a commutative diagram with exact rows.
the Mayer–Vietoris sequence is natural, so the inclusion maps \(C \hookrightarrow A\) and \(D \hookrightarrow B\) induce a commutative diagram of the two Mayer–Vietoris sequences.
3. Case \(A = B\): then \(X = A\) and \(Y = C \cup D\), and the diagram has \(H_n(A) = H_n(B)\) with the identity map (an isomorphism) on every third term.
when \(A = B\), the two “halves” coincide, so the vertical maps on the \(A\)-terms are identities.
4. Applying the algebraic lemma (<1>1) yields the relative Mayer–Vietoris sequence \[\cdots \to H_{n+1}(X, Y) \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X, Y) \to \cdots\] (equivalently, the long exact sequence of the pair \((X, Y)\)).
1 and 3.
5. Case \(C = D\): similarly, the vertical maps on the \(C\)-terms are identities, and the lemma yields the relative Mayer–Vietoris sequence for the pair.
the same argument with \(C = D\).
6. Q.E.D.
4 and 5.