The integral cohomology ring of symmetric products of CW-complexes and topology of symmetric products of Riemann surfaces

It is shown that 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 $H^*(X;\mathbb{Z})/\mathrm{Tor}$ (Theorem 3.1). Moreover, an explicit description of this functor is given. It is also considered an important particular case, when $X$ is a compact Riemann surface $M^2_g$ of genus $g$. The celebrated Macdonald Theorem of 1962 gives an explicit description of the integral cohomology ring $H^*(\mathrm{Sym}^n M^2_g;\mathbb{Z})$. A rigorous analysis of the original Macdonald's proof shows, that it contains three gaps. All these gaps have been eliminated by Seroul in 1972 and, therefore, Seroul obtained a correct proof of the Macdonald Theorem. Nevertheless, in the unstable case $2\le n\le 2g-2$, in the statement of the Macdonald Theorem there is a subitem, that needs a correction even for rational cohomology rings (Theorem 4.1). In the paper the following famous conjecture is proved. Let us denote by $M^2_{g,k}$ an arbitrary compact Riemann surface of genus $g\ge 0$ with $k\ge 1$ punctures. Conjecture (Blagojević-Grujić-Živaljević, 2003). Let us take numbers $n\ge 2, g, g'\ge 0, k,k'\ge 1,$ with conditions $2g+k=2g'+k'$ and $g\ne g'$. Then homotopy equivalent open manifolds $\mathrm{Sym}^n M^2_{g,k}$ and $\mathrm{Sym}^n M^2_{g',k'}$ are not homeomorphic.