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

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.