Planar non-formality of the little discs operad in characteristic two

TL;DR

This paper demonstrates that the little discs operad $D_2$ is not formal over the field with two elements when considered as a planar operad, using explicit homological obstruction calculations.

Contribution

It provides the first explicit computation showing non-formality of $D_2$ over $F_2$ in the planar setting, utilizing the spineless cacti operad model.

Findings

$D_2$ is not formal over $F_2$ as a planar operad

Explicit homological obstruction computed

Uses spineless cacti operad as chain model

Abstract

We show that the little discs operad $D_2$ is not formal over $\mathbb{F}_2$ as a planar (or non-symmetric) operad. We compute explicitly the homological obstruction using as chain model the cells of the spineless cacti operad