Arrangements of symmetric products of spaces

Using the topological technique of diagrams of spaces, we calculate the homology of the union and the complement of finite arrangements of subspaces of the form $D+S{P}^{n-d}(X)$ in symmetric products $S{P}^n(X)$ where $D\in SP^d(X)$. As an application we include a computation of the homology of the homotopy end space of the open manifold $S{P}^n(M_{g,k})$, where $M_{g,k}$ is a Riemann surface of genus $g$ punctured at $k$ points, a problem which was originally motivated by the study of commutative $(m+k,m)$-groups.