On the calculation of the cohomology of the third finite subset space of spheres

TL;DR

This paper computes the mod 2 cohomology groups of the third finite subset space of spheres, expanding understanding of their topological properties using symmetric product cohomology results.

Contribution

It introduces a method to calculate the cohomology of the third finite subset space of spheres leveraging existing symmetric product cohomology results.

Findings

Cohomology groups of the third finite subset space of spheres are explicitly computed.

The approach connects finite subset spaces with symmetric product cohomology.

Results enhance understanding of the topology of finite subset spaces.

Abstract

In this paper we provide a computation of the mod 2 cohomology groups of the third finite subset space of the sphere $S^n$ using known results about the cohomology of the symmetric product of spheres.