For \(SX\) the suspension of \(X\), show by a Mayer–Vietoris sequence that there are isomorphisms \(\tilde{H}_n(SX) \approx \tilde{H}_{n-1}(X)\) for all \(n\).
Solution
Write the suspension as the union of the upper and lower cones, \[ SX=C_+X\cup C_-X. \] Choose slightly enlarged open cone neighborhoods so that Mayer–Vietoris applies. Each cone is contractible, while their intersection deformation retracts onto the equatorial copy of \(X\).
The reduced Mayer–Vietoris sequence contains \[ \widetilde H_n(C_+X)\oplus\widetilde H_n(C_-X) \longrightarrow \widetilde H_n(SX) \xrightarrow{\delta} \widetilde H_{n-1}(X) \longrightarrow \widetilde H_{n-1}(C_+X)\oplus\widetilde H_{n-1}(C_-X). \] Both cone terms vanish, so the connecting homomorphism is an isomorphism: \[ \boxed{ \widetilde H_n(SX)\cong\widetilde H_{n-1}(X)} \] for every \(n\).