Symmetric products of surfaces and the cycle index

We study some of the combinatorial structures related to the signature ofG-symmetric products of (open) surfacesSP _G ^m (M)=M ^m/G whereG ⊂S _m.The attention is focused on the question, what information about a surfaceM can be recovered from a symmetric productSP ⁿ(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(SP _G ^m (M))in terms of the cycle index ofG, 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 surfacesM _g,k,M _g′,k′such that the manifoldsSP ^m(M _g,k)andSP ^m(M)_g′,k′)are often not homeomorphic, although they always have the same homotopy type provided 2 _g +k=2 _g′ +k′ andk,k′≥1. Supported by the Serbian Ministry for Science and Technology, Grant No. 1643.