I suppose my question has two parts. To begin Tillmann defines the operad $\mathcal{M}$ by its groupoids: $$\coprod_{g\geq 0} B\mathcal{S}_{g,n,1}$$ In the paper she proves that algebras over this operad are infinite loop spaces. The fact this operad is, in a way, graded by genus gives me the impression that the $g\leq n$ genus components determine the $n$-fold loop spaces. Similar to how we define an $E_{\infty}$-operad by the colimit of the $E_n$-algebras.
My question then would be what role is the genus playing here and how? If I restrict to the genus $0$ case can I use this operad to define $A_{\infty}/E_1$-algebras? If I restrict to the $g\leq n$ case do I get $E_n$-algebras? If so I am failing to see how the genus of the surface gives me the homotopy coherent commutativity that I seek.