The cohomology ring away from 2 of configuration spaces on real projective spaces

TL;DR

This paper computes the cohomology ring of configuration spaces in real projective spaces using spectral sequences and orbit space analysis, revealing new algebraic and topological properties.

Contribution

It introduces a novel method to compute the cohomology ring of configuration spaces in real projective spaces and analyzes their topological complexities.

Findings

Cohomology ring computed for configuration spaces in real projective spaces.

Spectral sequence collapses after first differential for odd dimensions.

Topological complexities of orbit configuration spaces determined.

Abstract

Let R be a commutative ring containing 1/2. We compute the R-cohomology ring of the configuration space F(m,k) of k ordered points in the m-dimensional real projective space. The method uses the observation that the orbit configuration space of k ordered points in the m-dimensional sphere (with respect to the antipodal action) is a 2^k-fold covering of F(m,k). This implies that, for odd m, the Leray spectral sequence for the inclusion of F(m,k) in the k-fold Cartesian self power of the m-dimensional real projective space collapses after its first non-trivial differential, just as it does when the projective space is replaced by a complex projective variety. The method also allows us to handle the R-cohomology ring of the configuration space of k ordered points in a punctured real projective space. Lastly, we compute the Lusternik-Schnirelmann category and all of the higher topological complexities of some of the auxiliary orbit configuration spaces.