On Homeomorphism Type of Symmetric Products of Compact Riemann Surfaces with Punctures

Abstract: Let $M^2_{g,k}$ and $M^2_{g',k'}$ be compact Riemann surfaces with punctures ($g,g'\ge 0$ - genuses, $k,k'\ge 1$ - number of punctures). For any Hausdorff space $X$ the quotient space $\mathrm{Sym}^nX := X^n/S_n$ is the $n$-th symmetric product of $X, \ n\ge 2$. It is well known, that $\mathrm{Sym}^n M^2_{g,k}$ is a smooth quasi-projective variety. Open manifolds $\mathrm{Sym}^n M^2_{g,k}$ and $\mathrm{Sym}^n M^2_{g',k'}$ are homotopy equivalent iff $\ 2g+k=2g'+k'$. Blagojevi\'{c}-Gruji\'{c}-\v{Z}ivaljevi\'{c} Conjecture (2003). Fix any $n\ge 2$, and two pairs $(g,k)$ and $(g',k')$ with the condition $2g+k=2g'+k'$. If $g\ne g'$, then open manifolds $\mathrm{Sym}^n M^2_{g,k}$ and $\mathrm{Sym}^n M^2_{g',k'}$ are not continuously homeomorphic. The conjecture was proved in 2003 in the paper by P.Blagojevi\'{c}, V.Gruji\'{c} and R.\v{Z}ivaljevi\'{c} for the case $\mathrm{max}(g,g') \ge \frac{n}{2}$ (this implies the case $n=2$). As far as the author knows, up to this moment there were no results if $\mathrm{max}(g,g') < \frac{n}{2}$. The aim of this paper is to prove the conjecture in full generality.