Two responses to symmetric power question..........DMD ___________________________________ Subject: Reply symm.pwrs Date: Thu, 24 May 2001 20:15:10 +0200 (MET DST) From: Sadok Kallel Here's a quick answer to the question on symmetric products. Best, - Sadok. ________________________ > 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 the closed oriented case, the cohomology is entirely worked out in MacDonald's topology paper: Symmetric products of an algebraic curve. Topology 1 1962 319--343. for some more geometry see perhaps S. Kallel, ``Divisor spaces on punctured Riemann surfaces'', Trans. Am. Math. Soc. {\bf 350} (1998), 135--164. Still in that case, the SP^n(M) are analytic fibrations over a complex torus (the Jacobian of M) when n sufficiently large. Much information (for all n) can be found in the book of Arbarello, Cornalba, Griffiths and Harris: "Geometry of Algebraic Curves Volume I" (in fact most of the book can be thought of as the study of the geometry of these spaces). This manifold (and other variants) has been of great use in recent work on the topology of certain mapping spaces and other configuration spaces. Of possible interest is S. Kallel, R.J. Milgram ``The geometry of spaces of holomorphic maps from a Riemann surface into complex projective space'', J. Diff. Geometry {\bf 47} (1997) 321--375. and references therein. __________________________________ Subject: Re: symm pwr query & Paris in 2004 Date: Thu, 24 May 2001 20:48:23 -0500 From: Clarence Wilkerson Regarding sign. of symmetric powers of manifolds, I seem to recall a book by Hirzebruch and Zagier, in 1973 or so in the Publish or Perish series that works out material of this type. Clarence Wilkerson