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

TL;DR

The paper resolves the Blagojević-Grujić-Živaljević conjecture in full generality, showing that for fixed $n\ge2$ and pairs $(g,k),(g',k')$ with $2g+k=2g'+k'$, the open manifolds $\mathrm{Sym}^n M^2_{g,k} \times \mathbb{R}^N$ and $\mathrm{Sym}^n M^2_{g',k'} \times \mathbb{R}^N$ are not homeomorphic when $g\neq g'$, even beyond the previously known range. The argument combines a detailed study of the cohomology rings (including integral and mod $2$ structures), the first Chern class via Macdonald’s formula, and the relation between Chern and Stiefel-Whitney classes to produce a robust invariant $w_2$ that encodes genus data after stabilization. A locality/doubling approach is developed to address open-manifold invariance questions for $w_2$ under purely continuous homeomorphisms, complemented by results on Pontryagin classes vanishing and a discussion of compactifications by attaching a boundary; the boundary is shown to be smoothable for $n\ge3$ and a conjecture on non-homotopy-equivalence of boundary manifolds is proposed. These pieces collectively advance the classification of symmetric products of punctured Riemann surfaces beyond homotopy type and into refined topological type, with potential implications for smoothable compactifications and boundary invariants.

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 $\ 2g+k=2g'+k'$. Blagojević-Grujić-Živaljević 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.Blagojević, V.Grujić and R.Živaljević 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.