Use the Mayer–Vietoris sequence to show there are isomorphisms \(\tilde{H}_n(X \vee Y) \approx \tilde{H}_n(X) \oplus \tilde{H}_n(Y)\) if the basepoints of \(X\) and \(Y\) that are identified in \(X \vee Y\) are deformation retracts of neighborhoods \(U \subset X\) and \(V \subset Y\).
Solution
Let \[ W=X\vee Y \] with common basepoint \(*\). By hypothesis there are neighborhoods \(U\subset X\) and \(V\subset Y\) that deformation retract to \(*\).
Choose subspaces of \(W\) \[ A=X\cup V, \qquad B=U\cup Y, \] slightly thickened if necessary so that they form an excisive cover for Mayer–Vietoris.
1. The inclusions induce homotopy equivalences\[ A\simeq X, \qquad B\simeq Y, \qquad A\cap B=U\cup V\simeq *. \]
Retract \(V\) to the wedge point while fixing \(X\), and similarly retract \(U\) while fixing \(Y\). Since both \(U\) and \(V\) retract to the common basepoint, their wedge \(U\cup V\) is contractible.
2. The reduced Mayer–Vietoris sequence gives\[ \widetilde H_n(W)\cong\widetilde H_n(A)\oplus\widetilde H_n(B) \]for every \(n\).
The reduced Mayer–Vietoris sequence contains \[ \widetilde H_n(A\cap B) \longrightarrow \widetilde H_n(A)\oplus\widetilde H_n(B) \longrightarrow \widetilde H_n(W) \longrightarrow \widetilde H_{n-1}(A\cap B). \] By 1 both outer groups vanish, so the middle arrow is an isomorphism. Replacing \(A\) and \(B\) by their deformation retracts yields \[ \boxed{ \widetilde H_n(X\vee Y)\cong \widetilde H_n(X)\oplus\widetilde H_n(Y).} \]