Lemma 3.4.6. Let \((h^*,\sigma )\) be a reduced cohomology theory. For each \(n\in \Z \), the restriction \[ h^n\colon \Ho (\An _{*,\geq 1})\catop \longrightarrow \Ab \longrightarrow \Set \] satisfies the Mayer–Vietoris property.
Proof. Consider a pushout square of connected pointed animae
The pushout property gives an isomorphism \(\cofib (k)\iso \cofib (j)\). The long exact cohomology sequences of Exercise 3.1.3 therefore fit into a commutative diagram
Suppose that \(\alpha \in h^n(A)\) and \(\beta \in h^n(B)\) satisfy \(k^*\alpha =l^*\beta \). The boundary of \(\beta \) vanishes, since its image under the right-hand isomorphism is the boundary of \(l^*\beta =k^*\alpha \). Hence \(\beta \) lifts to some \(\gamma _0\in h^n(X)\). The element \(\alpha -i^*\gamma _0\) lies in the kernel of \(k^*\), so it comes from \(h^n(\cofib (k))\). Transporting a preimage across the left-hand isomorphism and then mapping it to \(h^n(X)\) gives an element \(\gamma _1\) whose restriction to \(B\) vanishes and whose restriction to \(A\) is \(\alpha -i^*\gamma _0\). Thus \(\gamma _0+\gamma _1\) restricts to \((\alpha ,\beta )\), proving the required surjectivity. □
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