On Integral Cohomology Ring of Symmetric Products

TL;DR

This paper characterizes the integral cohomology ring of symmetric products of CW-complexes, providing an explicit functorial description and addressing gaps in Macdonald's classical theorem for Riemann surfaces.

Contribution

It offers a functorial description of the cohomology ring modulo torsion for symmetric products and corrects aspects of Macdonald's theorem in the unstable case.

Findings

The cohomology ring modulo torsion is a functor of the base space's cohomology.

Explicit description of the cohomology ring for symmetric products of CW-complexes.

Identification and correction of gaps in Macdonald's theorem for certain cases.

Abstract

We prove that the integral cohomology ring modulo torsion $H^*(\mathrm{Sym}^n X;\mathbb{Z})/\mathrm{Tor}$ for the symmetric product of a connected CW-complex $X$ of finite homology type is a functor of $H^*(X;\mathbb{Z})/\mathrm{Tor}$ (see Theorem 1). Moreover, we give an explicit description of this functor. We also consider the important particular case when $X$ is a compact Riemann surface $M^2_g$ of genus $g$. There is a famous theorem of Macdonald of 1962, which gives an explicit description of the integral cohomology ring $H^*(\mathrm{Sym}^n M^2_g;\mathbb{Z})$. The analysis of the original proof by Macdonald shows that it contains three gaps. All these gaps were filled in by Seroul in 1972, and, therefore, he obtained a complete proof of Macdonald's theorem. Nevertheless, in the unstable case $2\le n\le 2g-2$ Macdonald's theorem has a subsection, that needs a slight correction even over $\mathbb{Q}$ (see Theorem 2).