The Integral Cohomology Ring of Symmetric Products of CW Complexes and Topology of Symmetric Products of Riemann Surfaces

D. V. Gugnin

Informatics and Automation, Topology, Geometry, Combinatorics, and Mathematical Physics, Tome 326 (2024), pp. 148-172

Résumé

We show that the integral cohomology ring modulo torsion $H^*(\mathrm {Sym}^n X;\mathbb {Z})/\mathrm {Tor}$ for symmetric products of connected countable CW complexes of finite homology type is a functor of the ring $H^*(X;\mathbb {Z})/\mathrm {Tor}$, and we give an explicit description of this functor. There is an important particular case of this situation with $X$ a compact Riemann surface $M^2_g$ of genus $g$. Macdonald's famous theorem of 1962 provides an explicit description of the integral cohomology ring $H^*(\mathrm {Sym}^n M^2_g;\mathbb {Z})$. However, a careful analysis of Macdonald's original proof shows that it has three gaps. All these gaps were filled by Seroul in 1972, and thus Seroul obtained a complete proof of Macdonald's theorem. Nevertheless, in the unstable case $2\le n\le 2g-2$ there is a subclause of Macdonald's theorem that needs to be corrected even for rational cohomology rings. In the paper we prove the following well-known conjecture (Blagojević–Grujić–Živaljević, 2003): Denote by $M^2_{g,k}$ an arbitrary compact Riemann surface of genus $g\ge 0$ with $k\ge 1$ punctures. Take numbers $n\ge 2$, $g,g'\ge 0$, and $k,k'\ge 1$ such that $2g+k=2g'+k'$ and $g\ne g'$. Then the homotopy equivalent open manifolds $\mathrm {Sym}^n M^2_{g,k}$ and $\mathrm {Sym}^n M^2_{g',k'}$ are not homeomorphic.

Détail

Keywords: symmetric products, Riemann surfaces, integral cohomology, characteristic classes.

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