Abstract
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.
