Formality of the framed little 2-discs operad and semidirect products

TL;DR

This paper proves the formality of the framed little 2-discs operad by extending Kontsevich's chain of quasi-isomorphisms to include circle actions, demonstrating compatibility with the semidirect product construction.

Contribution

It establishes the formality of the framed little 2-discs operad by adapting existing formality results to the context of semidirect products with circle actions.

Findings

Proved the formality of the framed little 2-discs operad.

Extended Kontsevich's quasi-isomorphisms to include circle actions.

Showed compatibility of chain complexes with semidirect products.

Abstract

We prove that the operad of framed little 2-discs is formal. Tamarkin and Kontsevich each proved that the unframed 2-discs operad is formal. The unframed 2-discs is an operad in the category of S^1-spaces, and the framed 2-discs operad can be constructed from the unframed 2-discs by forming the operadic semidirect product with the circle group. The idea of our proof is to show that Kontsevich's chain of quasi-isomorphisms is compatible with the circle actions and so one can essentially take the operadic semidirect product with the homology of S^1 everywhere to obtain a chain of quasi-isomorphisms between the homology and the chains of the framed 2-discs.