Formality of the framed little 2-discs operad and semidirect products - Jeffrey Giansiracusa (Contemporary Mathematics, 2010)
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.In Homotopy theory of function spaces and related topics, 115–121, Contemp. Math., 519, Amer. Math. Soc., Providence, RI, 2010. arXiv:0911.4428