The framed little 2-discs operad and diffeomorphisms of handlebodies

TL;DR

This paper establishes homotopy equivalences between derived modular envelopes of certain cyclic operads and classifying spaces of diffeomorphism groups of handlebodies and related structures, providing new proofs and extending known results.

Contribution

It introduces a new approach to relate cyclic operads to classifying spaces of diffeomorphism groups, offering elementary proofs of key theorems and extending their scope.

Findings

Homotopy equivalence between the derived modular envelope of the framed little 2-discs operad and handlebody diffeomorphism classifying spaces.

Elementary proof of Costello's theorem relating associative operad's modular envelope to moduli spaces of Riemann surfaces.

Recovery of Braun's theorem connecting cyclic operads with involution to moduli spaces of Klein surfaces.

Abstract

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.