The Deligne-Mumford operad as a trivialization of the circle action

We prove that the tree-like Deligne-Mumford operad is a homotopical model for the trivialization of the circle in the higher-genus framed little discs operad. Our proof is based on a geometric argument involving nodal annuli. We use Riemann surfaces with analytically parametrized boundary as a model for higher-genus framed little discs.