The integral cohomology of configuration spaces of pairs of points in real projective spaces

TL;DR

This paper computes the integral cohomology rings of configuration spaces of two points in real projective spaces, revealing torsion structures and advancing the understanding of their topological complexity.

Contribution

It provides the first complete calculation of the integral cohomology of these configuration spaces, including torsion details, and applies this to determine symmetric topological complexity.

Findings

Cohomology rings are quotients of known groups' cohomology.

Configuration spaces have only 2- and 4-torsion.

Complete the computation of symmetric topological complexity for certain real projective spaces.

Abstract

We compute the integral cohomology ring of configuration spaces of two points on a given real projective space. Apart from an integral class, the resulting ring is a quotient of the known integral cohomology of the dihedral group of order 8 (in the case of unordered configurations, thus has only 2- and 4-torsion) or of the elementary abelian 2-group of rank 2 (in the case of ordered configurations, thus has only 2-torsion). As an application, we complete the computation of the symmetric topological complexity of real projective spaces of dimensions of the form 2^i+j for non-negative i and j with j<3.