Homology of a wedge sum by Mayer--Vietoris

Assume that the wedge point has an open neighborhood in \(X\) and one in \(Y\) that deformation retract onto it; this holds when \(X\) and \(Y\) are CW complexes wedged at \(0\)-cells. Choose open neighborhoods \(U,V\subset X\vee Y\) deformation retracting onto \(X\) and \(Y\), respectively, with \(U\cap V\) contractible. The reduced Mayer–Vietoris sequence then has, for every \(n\), \[ 0=\widetilde H_n(U\cap V) \longrightarrow \widetilde H_n(U)\oplus\widetilde H_n(V) \longrightarrow \widetilde H_n(X\vee Y) \longrightarrow \widetilde H_{n-1}(U\cap V)=0. \] Hence \[ \boxed{\widetilde H_n(X\vee Y) \cong \widetilde H_n(X)\oplus\widetilde H_n(Y)} \] for all \(n\). In particular, when \(X\) and \(Y\) are path connected, \[ H_0(X\vee Y)\cong\mathbb Z, \qquad H_n(X\vee Y)\cong H_n(X)\oplus H_n(Y)\quad(n>0). \]