  The framed little 2-discs operad is homotopy equivalent to a cyclic operad.
We show that the derived modular envelope of this cyclic operad (i.e., the
modular operad freely generated in a homotopy invariant sense) is homotopy
equivalent to the modular operad made from classifying spaces of diffeomorphism
groups of 3-dimensional handlebodies with marked discs on their boundaries. A
modification of the argument provides a new and elementary proof of K.
Costello's theorem that the derived modular envelope of the associative operad
is homotopy equivalent to the ``open string'' modular operad made from moduli
spaces of Riemann surfaces with marked intervals on the boundary. Our technique
also recovers a theorem of C. Braun that the derived modular envelope of the
cyclic operad that describes associative algebras with involution is homotopy
equivalent to the modular operad made from moduli spaces of unoriented Klein
surfaces with open string gluing.
