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\begin{document}

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\section*{framed little 2-disk operad}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{framed little 2-disk operad} is the [[operad]] $fD_2$ in [[Top]] whose [[topological space]] $fD_2(n)$ of $n$-ary operations is the space of maps

\begin{displaymath}
\coprod_n D \to D
\end{displaymath}
from $n$-copies of the 2-[[ball]] to itself, which restrict on each component to a map that is a combination of

\begin{itemize}%
\item a translation


\item a dilatation


\item a rotation



\end{itemize}
of the disk (regarded via its standard [[embedding]] $D \hookrightarrow \mathbb{R}^2$ into the 2-simensional [[Cartesian space]]) such that the images of all disks are disjoint.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This differs from the [[little 2-disk operad]] by the fact that rotations of the disks are admitted. Under passing to chains and then to [[homology]], this operation gives rise to the BV-operator in a [[BV-algebra]]. See \hyperlink{Properties}{Properties} below.

\end{remark}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\begin{theorem}
\label{}\hypertarget{}{}
The [[homology]] of the framed little 2-disk operad in chain complexes is the [[BV-operad]] $BV$ the operad for [[BV-algebra]]s:

\begin{displaymath}
BV \simeq H_\bullet(fD_2)
  \,.
\end{displaymath}
\end{theorem}
This is due to (\hyperlink{Getzler}{Getzler}).

\begin{theorem}
\label{}\hypertarget{}{}
The framed little disk operad is [[formal dg-algebra|formal]] in characteristic zero.

This means that there is a zig-zag of [[quasi-isomorphism]]s

\begin{displaymath}
C_\bullet(fD_2) 
   \stackrel{\simeq}{\leftarrow}
   \stackrel{\simeq}{\to}
   \cdots 
   \stackrel{\simeq}{\leftarrow}
   \stackrel{\simeq}{\to}
   H_\bullet(fD_2)
  \,.
\end{displaymath}
\end{theorem}
This is due to \hyperlink{Severa09}{Severa 09}, \hyperlink{GiansiracusaSalvatore09}{Giansiracusa-Salvatore 09)}. See also (\hyperlink{Valette}{Valette, slide 35}).

Accordingly one makes the following definition:

\begin{defn}
\label{}\hypertarget{}{}
The operad for [[homotopy BV-algebra]]s is any cofibrant [[resolution]] of $BV \simeq  H_\bullet(fD_2)$, or equivalently of $C_\bullet(fD_2)$.

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
Write $R \beta_j$ for the [[ribbon braid group]] on $j$ elements and $P R \beta_j$ for the [[kernel]] of the surjection $R \beta_j \to \Sigma_j$ onto the [[symmetric group]].

Say that a [[ribbon operad]] $P$ is an \textbf{$R_\infty$-operad} if the [[ribbon braid group]]s act freely and properly on $P$ and if each [[topological space]] $P(k)$ is [[contractible]].

\end{defn}
\begin{theorem}
\label{}\hypertarget{}{}
If $P$ is an $R_\infty$-operad, then the sequence of quotient spaces $\{P(n)/P R \beta_n\}$ forms a symmetric operad equivalent to the frame little disks operad.

\end{theorem}
This is (\hyperlink{Wahl}{Wahl, lemma 1.5.17}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item \textbf{framed little 2-disk operad}


\item [[framed little n-disk operad]]


\item [[BD operad]]



\end{itemize}
[[!include deformation quantization - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

The framed little 2-disk operad was introduced in

\begin{itemize}%
\item [[Ezra Getzler]], \emph{Batalin-Vilkovisky algebras and two-dimensional topological field theories} , Comm. Math. Phys. 159 (1994), no. 2, 265--285. (\href{http://arxiv.org/abs/hep-th/9212043}{arXiv})

\end{itemize}
For the relation to ribbons see

\begin{itemize}%
\item [[Nathalie Wahl]], \emph{Ribbon braids and related operads} PhD thesis, Oxford (2001) (\href{http://eprints.maths.ox.ac.uk/43/1/wahl.pdf}{pdf}).

\end{itemize}
The formality of $fD_2$ was shown in

\begin{itemize}%
\item [[Jeffrey Giansiracusa]], [[Paolo Salvatore]], \emph{Formality of the framed little 2-discs operad and semidirect products} , in: \emph{Homotopy theory of function spaces and related topics}, Cont. Math. 519, AMS, pp. 115-121 (\href{http://arxiv.org/PS_cache/arxiv/abs/0911/0911.4428}{arxiv 0911.4428})

\end{itemize}
and

\begin{itemize}%
\item [[Pavol Severa]], \emph{Formality of the chain operad of framed little disks}, (\href{http://arxiv.org/PS_cache/arxiv/abs/0902/0902.3576}{arxiv AT 0902.3576})

\end{itemize}
Discussion of [[homotopy BV-algebra]]s is in

\begin{itemize}%
\item [[Imma Galvez-Carrillo]], [[Andy Tonks]], [[Bruno Valette]], \emph{Homotopy Batalin-Vilkovisky algebras} (\href{http://arxiv.org/abs/0907.2246}{arXiv:0907.2246})

\end{itemize}
see also

\begin{itemize}%
\item [[Imma Galvez-Carrillo]], Vassily Gorbounov, [[Andy Tonks]], \emph{Homotopy Gerstenhaber structures and vertex algebras}, \href{http://arxiv.org/abs/math/0611231}{math/0611231.QA}

\end{itemize}
Slides of a talk summarizing this are at

\begin{itemize}%
\item [[Bruno Valette]], \emph{Homotopy Batalin-Vilkovisky algebras} (\href{http://math.unice.fr/~brunov/download/Homotopy%20BV.pdf}{pdf})

\end{itemize}
[[!redirects E2-operad]] [[!redirects E2-operads]]

[[!redirects E2 operad]] [[!redirects E2 operads]]

[[!redirects framed little disk operad]] [[!redirects framed little disks operad]]

[[!redirects framed little disk operads]] [[!redirects framed little disks operads]]

[[!redirects framed little disc operad]] [[!redirects framed little discs operad]]

[[!redirects framed little disc operads]] [[!redirects framed little discs operads]]

[[!redirects BV-operad]]



\end{document}