On Homeomorphism Type of Symmetric Products of Compact Riemann Surfaces with Punctures
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| Formato: | Preprint |
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2016
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| _version_ | 1866916659633061888 |
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| author | Gugnin, Dmitry V. |
| author_facet | Gugnin, Dmitry V. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1606_00453 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | On Homeomorphism Type of Symmetric Products of Compact Riemann Surfaces with Punctures Gugnin, Dmitry V. Algebraic Topology Algebraic Geometry Geometric Topology 55S15, 57R19, 57R20 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. |
| title | On Homeomorphism Type of Symmetric Products of Compact Riemann Surfaces with Punctures |
| topic | Algebraic Topology Algebraic Geometry Geometric Topology 55S15, 57R19, 57R20 |
| url | https://arxiv.org/abs/1606.00453 |