# On the unordered configuration space C(RP^n,2)

## TL;DR

This paper investigates the embedding and immersion properties of the unordered configuration space of two points in real projective space, providing optimal bounds and cohomological insights, especially when n is a power of two.

## Contribution

It establishes new optimal bounds for immersions and embeddings of C(RP^n,2) and offers a novel description of the mod-2 cohomology algebra of G_{n+1,2}.

## Key findings

- C(RP^n,2) cannot be immersed in R^{4n-2} when n is a 2-power.
- C(RP^n,2) can be immersed in R^{4n-3} when n is not a 2-power.
- Cohomological lower bounds for topological complexity are nearly optimal for 2-power n.

## Abstract

We prove that, if n is a 2-power, the unordered configuration space C(RP^n,2) cannot be immersed in R^{4n-2} nor embedded as a closed subspace of R^{4n-1}, optimal results, while if n is not a 2-power, C(RP^n,2) can be immersed in R^{4n-3}. We also obtain cohomological lower bounds for the topological complexity of C(RP^n,2), which are nearly optimal when n is a 2-power. We also give a new description of the mod-2 cohomology algebra of the Grassmann manifold G_{n+1,2}.