Subject: Question for the group
Date: Thu, 24 May 2001 17:23:37 +0200 (CEST)
From: Rade Zivaljevic <rade@turing.mi.sanu.ac.yu>
To: dmd1@lehigh.edu

 Dear prof. Davis,
 Thanks for the energy invested in maintaining
 the discussion group!
 I think this is a great idea. Rade

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Question for the topology discussion group

(Rade \v Zivaljevi\' c; Belgrade)

Symmetric powers of Riemann (and other) surfaces.

   The signature $Sign(SP^{2n}(M_g))$ is a binomial coefficient,
   $(-1)^n{g-1\choose n}$.

   This result ought to be well known.
   Q_1: What are the references to this result?

   We have recently calculated, as part of our Topology/Combinatorics
   interaction seminar in Belgrade, the signature of the rational homology
   manifold $SP^{n}_G(M_g) := ((M_g)^n)/G$ as a function of both
   $g$ and $G\subset S_n$. Our answer is expressed in terms of the cycle
   index $Z(G;x_1,\ldots , x_n)$, a polynomial which appears in
   Polya enumeration theory and includes the answer above as a
   special case via a well known combinatorial identity.

   Q_2: Are there references to the signature of $SP^{n}_G(M_g)$ as a
   function of both $G$ and $g$.

   Rade Zivaljevic

   G - T - A seminar, Belgrade

  p.s. What is generally known about the manifold $SP^n(M):= M^n/S_n$ where $M$
  is a closed (open), orientable or nonorientable surface?

  For example the homeomorphism $SP^n(RP^2)\cong RP^{2n}$ (proved originally by
  Dupont and Lusztig ?!), is ``attributed'' to Maxwell in a charming paper of V. Arnold:
  Topological content of the Maxwell theorem on multiple representations
  of spherical functions, in Topological Methods in Nonlinear Analysis;
  J. Juliusz Shouder Center,  7 (1996), 205 - 217.