On Integral Cohomology Ring of Symmetric Products
Dmitry V. Gugnin

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 for the symmetric product of a connected CW-complex of finite homology type is a functor of (see Theorem 1). Moreover, we give an explicit description of this functor. We also consider the important particular case when is a compact Riemann surface of genus . There is a famous theorem of Macdonald of 1962, which gives an explicit description of the integral cohomology ring . 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 Macdonald's theorem has a subsection, that needs a slight correction…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology
