# On Integral Cohomology Ring of Symmetric Products

Research paper by Dmitry V. Gugnin

Indexed on: 06 Feb '15Published on: 06 Feb '15Published in: Mathematics - Algebraic Topology

#### Abstract

We prove that the integral cohomology ring modulo torsion $H^*(\mathrm{Sym}^n X;\mathbb{Z})/\mathrm{Tor}$ for symmetric products of connected CW-complexes $X$ of finite homology type is a functor of $H^*(X;\mathbb{Z})/\mathrm{Tor}$. Moreover, we give an explicit description of this functor. Also we apply our knowledge to the case when $X$ is a compact Riemann surface $M^2_g$ of arbitrary genus $g\ge 0$. Here we verify the famous theorem of I.G.Macdonald of 1962, which gives an explicit determination of the integral cohomology ring $H^*(\mathrm{Sym}^n M^2_g;\mathbb{Z})$.