Homotopically trivializing the circle in the framed little disks

TL;DR

This paper establishes a deep equivalence between trivializing the circle action in the framed little disks operad and the action of the Deligne-Mumford-Knudsen operad, confirming Kontsevich's suggestion in a derived sense.

Contribution

It proves that trivializing the circle in the framed little disks operad is equivalent to an action of the Deligne-Mumford-Knudsen operad, refining previous results.

Findings

Equivalence of circle trivialization and Deligne-Mumford-Knudsen operad action

Improvement over earlier results by the author and Vallette

Confirmation of Kontsevich's suggestion in a derived context

Abstract

This paper confirms the following suggestion of Kontsevich. In the appropriate derived sense, an action of the framed little disks operad and a trivialization of the circle action is the same information as an action of the Deligne-Mumford-Knudsen operad. This improves an earlier result of the author and Bruno Vallette.