Symmetric products of surfaces and the cycle index

We study some of the combinatorial structures related to the signature of G-symmetric products of (open) surfaces SPGm(M) = M m/G where G ⊂ Sm. The attention is focused on the question, what information about a surface M can be recovered from a symmetric product SPn(M). The problem is motivated in part by the study of locally Euclidean topological commutative (m + k, m)-groups, [16]. Emphasizing a combinatorial point of view we express the signature Sign(SPG m(M)) in terms of the cycle index Z(G; x̄) of G, a polynomial which originally appeared in Pólya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfaces Mg,k, Mg,k, such that the manifolds SPm(Mg,k) and SPm(Mg,k) are often not homeomorphic, although they always have the same homotopy type provided 2g + k = 2g + k and k, k ≥ 1.