Non-formality of the odd dimensional framed little balls operads

TL;DR

This paper proves that the chain operad of the framed little balls operad is not formal over the rationals in odd dimensions greater than 4, revealing limitations in their algebraic simplifications.

Contribution

It demonstrates the non-formality of the chain operad for framed little balls in certain odd dimensions, a novel result in operad theory.

Findings

Chain operad is not formal over rationals in specified dimensions.

Non-formality holds for odd dimensions greater than 4.

Results impact understanding of algebraic structures in topology.

Abstract

We prove that the chain operad of the framed little balls (or disks) operad is not formal as a non-symmetric operad over the rationals if the dimension of their balls is odd and greater than 4.