Symmetric products of surfaces and the cycle index

2003/06/27 by Pavle Blagojevic, Pavle V. M. Blagojević, Vladimir Grujić +6
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #Mathematics and Applications #math.AT #math.CO #msc:14H40 #msc:55R80 #msc:55S15 #msc:57R19

paper · pdf · doi:10.48550/arxiv.math/0306397

arxiv created 2003/06/27 · arxiv updated 2009/11/30

Abstract

We express the signature \rm Sign(SP^m_G(M)) of the symmetric product SPⁿ(M) of an (open) surface M in terms of the cycle index Z(G; x) of G, a polynomial which originally appeared in P' olya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfaces M_g,k, M_g',k' such that the manifolds SP^m(M_g,k) and SP^m(M_g',k') are often not homeomorphic, although they always have the same homotopy type provided 2g+k = 2g'+k' and k,k'≥ 1.

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