On Integral Cohomology Ring of Symmetric Products
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2015
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| _version_ | 1866917965008470016 |
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| author | Gugnin, Dmitry V. |
| author_facet | Gugnin, Dmitry V. |
| contents | We prove that the integral cohomology ring modulo torsion $H^*(\mathrm{Sym}^n X;\mathbb{Z})/\mathrm{Tor}$ for the symmetric product of a connected CW-complex $X$ of finite homology type is a functor of $H^*(X;\mathbb{Z})/\mathrm{Tor}$ (see Theorem 1). Moreover, we give an explicit description of this functor.
We also consider the important particular case when $X$ is a compact Riemann surface $M^2_g$ of genus $g$. There is a famous theorem of Macdonald of 1962, which gives an explicit description of the integral cohomology ring $H^*(\mathrm{Sym}^n M^2_g;\mathbb{Z})$. The analysis of the original proof by Macdonald shows that it contains three gaps. All these gaps were filled in by Seroul in 1972, and, therefore, he obtained a complete proof of Macdonald's theorem.
Nevertheless, in the unstable case $2\le n\le 2g-2$ Macdonald's theorem has a subsection, that needs a slight correction even over $\mathbb{Q}$ (see Theorem 2). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1502_01862 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | On Integral Cohomology Ring of Symmetric Products Gugnin, Dmitry V. Algebraic Topology Algebraic Geometry 55S15, 57R19 We prove that the integral cohomology ring modulo torsion $H^*(\mathrm{Sym}^n X;\mathbb{Z})/\mathrm{Tor}$ for the symmetric product of a connected CW-complex $X$ of finite homology type is a functor of $H^*(X;\mathbb{Z})/\mathrm{Tor}$ (see Theorem 1). Moreover, we give an explicit description of this functor. We also consider the important particular case when $X$ is a compact Riemann surface $M^2_g$ of genus $g$. There is a famous theorem of Macdonald of 1962, which gives an explicit description of the integral cohomology ring $H^*(\mathrm{Sym}^n M^2_g;\mathbb{Z})$. The analysis of the original proof by Macdonald shows that it contains three gaps. All these gaps were filled in by Seroul in 1972, and, therefore, he obtained a complete proof of Macdonald's theorem. Nevertheless, in the unstable case $2\le n\le 2g-2$ Macdonald's theorem has a subsection, that needs a slight correction even over $\mathbb{Q}$ (see Theorem 2). |
| title | On Integral Cohomology Ring of Symmetric Products |
| topic | Algebraic Topology Algebraic Geometry 55S15, 57R19 |
| url | https://arxiv.org/abs/1502.01862 |