Arrangements of symmetric products of spaces

Abstract

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 $D\in S{P}^d(X)$. 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.

This is an updated version of the paper. In this version some results (Proposition 1.7., Theorem 1.8, Theorem 1.9, Theorem 1.11) are now reformulated in the greater generality (over integer coefficients). Moreover, we now interpret Theorems 1.8 and 1.11 as a generalization of classical Steenrod's theorem to the case symmetric products of (simple) diagrams of spaces