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

TL;DR

This paper proves a conjecture that classifies symmetric products of punctured Riemann surfaces up to homeomorphism, showing they are distinguished by genus when certain conditions are met.

Contribution

The paper completes the proof of the Blagojević-Grujić-divaljevi7c conjecture for all cases, including when the maximum genus is less than half of n.

Findings

Symmetric products are homeomorphic iff the genus and puncture counts satisfy specific conditions.

The conjecture holds for all genus and puncture configurations, extending previous partial results.

The classification depends on the genus when the relation 2g+k=2g'+k' is satisfied.

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.