Ribbon braids and related operads - Wahl, N (2001)
This thesis consists of two parts, both being concerned with operads related to the ribbon braid groups. In the first part, we define a notion of semidirect product for operads and use it to study the framed $n$-discs operad (the semidirect product $f\mathcal{D}_n=\mathcal{D}_n\rtimes SO(n)$ of the little $n$-discs operad with the special orthogonal group). This enables us to deduce properties of $f\mathcal{D}_n$ from the corresponding properties for $\mathcal{D}_n$. We prove an equivariant recognition principle saying that algebras over the framed $n$-discs operad are $n$-fold loop spaces on $SO(n)$-spaces. We also study the operations induced on homology, showing that an $H(f\mathcal{D}_n)$-algebra is a higher dimensional Batalin-Vilkovisky algebra with some additional operators when $n$ is even. Contrastingly, for $n$ odd, we show that the Gerstenhaber structure coming from the little $n$-discs does not give rise to a Batalin-Vilkovisky structure. We give a general construction of operads from families of groups. We then show that the operad obtained from the ribbon braid groups is equivalent to the framed 2-discs operad. It follows that the classifying spaces of ribbon braided monoidal categories are double loop spaces on $S^1$-spaces. The second part of this thesis is concerned with infinite loop space structures on the stable mapping class group. Two such structures were discovered by Tillmann. We show that they are equivalent, constructing a map between the spectra of deloops. We first construct an `almost map', i.e a map between simplicial spaces for which one of the simplicial identities is satisfied only up to homotopy. We show that there are higher homotopies and deduce the existence of a rectification. We then show that the rectification gives an equivalence of spectra.