# Non-formality of planar configuration spaces in characteristic two

## TL;DR

This paper demonstrates that the ordered configuration space of four or more points in the plane exhibits non-formality in its singular cochain algebra over characteristic two, using explicit obstruction classes.

## Contribution

It introduces a novel explicit obstruction class in Hochschild cohomology proving non-formality of certain configuration spaces in characteristic two.

## Key findings

- Configuration space of ≥4 points in the plane is non-formal over characteristic two.
- Configuration spaces with points ≤ dimension are intrinsically formal over any ring.
- Constructs explicit obstruction classes using the Barratt-Eccles-Smith model.

## Abstract

We prove that the ordered configuration space of 4 or more points in the plane has a non-formal singular cochain algebra in characteristic two. This is proved by constructing an explicit non trivial obstruction class in the Hochschild cohomology of the cohomology ring of the configuration space, by means of the Barratt-Eccles-Smith simplicial model. We also show that if the number of points does not exceed its dimension, then an euclidean configuration space is intrinsically formal over any ring.