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\begin{document}
\title{Deligne conjecture}
\author{James McClure}
\date{}
\maketitle
\section{Introduction}
Let $A$ be an associative ring. Then the Hochschild cochains are given by $C^p(A)=\Hom(A^{\otimes p},A)$, with differential
\begin{eqnarray*}
(d\varphi)(a_1\otimes \cdots \otimes a_{n+1}) &=& a_1\varphi(a_2\otimes \cdots \otimes a_{n+1}) \\ && + \sum_{i=1}^p (-1)^i \varphi(\cdots \ox a_i a_{i+1} \ox \cdots) \\ && + (-1)^{p+1} \varphi(a_1\ox\cdots\ox a_p)a_{p+1}.
\end{eqnarray*}
We then obtain the Hochschild homology $HH^*(A)=H^*(C^*(A))$. This comes with a cup product given by, for $\varphi\in HH^p(A)$ and $\psi\in HH^q(A)$, defining
\begin{equation*}
(\varphi\smile \psi)(a_1\ox\cdots \ox a_{p+q}) = \varphi(a_1\ox\cdots \ox a_p)\cdot \psi(a_{p+1}\ox\cdots\ox a_{p+q}).
\end{equation*}
This has no reason to be commutative at the cochain level, but amazingly it induces a commutative product on homology.
Gerstenhaber showed in 1962 that $HH^*A$ is a \textit{Gerstenhaber algebra}, meaning that it has a cup product and a compatible Lie bracket (the commutator of the star operation). Compatibility means that the cup product acts by derivations with respect to the Lie bracket.
Then, in 1973 Fred Cohen showed that if $X$ is a $\mc{D}_2$-algebra then $H_* X$ is a Gerstenhaber algebra.
Deligne noticed in 1993 that there was a connection between these two, and asked whether these came from the same place. More explicitly, he asked: \textbf{Does the Gerstenhaber structure on $HH^*A$ come from an action of an $E_2$ chain operad (i.e.\! a chain operad on the Hochschild cochains which is quasi-isomorphic to the singular chains $S_*\mc{D}_2$)?} The answer will be yes, although we'll actually normalize our cochains for technical reasons.
There are a number of proofs of this fact. The first was by Getzler-Jones in 1994, but it had a gap; their key ingredient was a cellular decomposition of the Fulton-MacPherson operad, but it wasn't cellular: they had lower cells attached to higher cells. This was fixed by Voronov in 2000. Meanwhile, in 1998 there were proofs in characteristic 0 by Tamarkin and Kontsevich, and then McClure-Smith obtained a proof over $\Z$ that even works for $A_\infty$ ring spectra. There have been many more proofs since then.
It turns out that McClure-Smith's method can be used to something quite a bit more general. One might ask: if one has a space and wants an operad to act, how do we get that to happen? The main one we know is that $\mc{D}_n$ acts on $\Omega^nX$ for a space $X$, and there are Segal $\Gamma$-spaces that admit $\infty$-actions, but what about the $\mc{D}_n$ operad acting on a general space? We will obtain a general condition that implies a $\mc{D}_n$-action.
McClure-Smith's work also leads to a combinatorial description of something quasi-isomorphic to the chains $S_* \mc{D}_2$.
A good reference for almost all this material is the survey paper by McClure-Smith, \textit{Operads and Cosimplicial Spaces: An Introduction} (on the arXiv). This includes a self-contained introduction to the $\mc{D}_n$ operads.
\section{Cosimplicial objects}
\begin{definition}
A \textit{cosimplicial object} in a category $\mc{A}$ is a covariant functor $\Delta\ra\mc{A}$. These can be described in terms of coface and codegeneracy maps.
\end{definition}
\begin{example}
The first example, which has $\mc{A}=\cat{Ab}$, is $C^*A$. This takes $\{0,\ldots, p\}$ to the Hochschild chains $C^p(A)$. We decompose the summands in the definition of $d:C^p(A)\ra C^{p+1}(A)$ into $d^0,\ldots,d^{p+1}$, and these are our codegeneracies.
\end{example}
\begin{example}
Let $S^*Y$ denote the singular cochains on a space $Y$. This takes $\{0,\ldots, p\}$ to $Map_{\cat{Set}}(S_pY,\Z)=Ma_{\cat{Top}}(\Delta^p,Y)$. This is the key motivating example.
\end{example}
We will mainly think about $\cat{Ab}$ and $\cat{Top}$, but in the background will be $Ch(R)$ and $\cat{Spectra}$.
\begin{definition}
The \textit{totalization} of a cosimplicial objects is given as follows. First, if $B^\bullet \in \cat{Ab}^\Delta$, then its \textit{conormalization} $N^*(B^\bullet)\in Ch^*$ is a chain complex defined by $N^p(B^\bullet) = \bigcap_{i=0}^{p-1} \ker(s^i)$ (where the $s^i$ are the codegeneracies). The differential is given by $d=\sum_{i=0}^{p+1} (-1)^i d^i$ (where the $d^i$ are the coface maps).
\end{definition}
\begin{example}
The normalization of the Hochschild chochain complex is $N^*(C^\bullet A)$ are the \textit{normalized} Hochschild cochains.
\end{example}
\begin{example}
$N^*(S^\bullet W)$ are the normalized singular cochains.
\end{example}
\subsubsection{Cosimplicial spaces}
\begin{example}
A key example of a cosimplicial space is $\Delta^\bullet$, which takes $\{1,\ldots, p\}$ to $\Delta^p$. This is not cofibrant, which makes life interesting in a number of ways.
\end{example}
\begin{example}
Note that a based space $Y$ is a lot like a coalgebra. The geometric cobar construction can be made by mimicking the usual cobar construction. Its totalization will be the loopspace of $Y$. We can also make the geometric cyclic cobar construction, and this will give the free loopspace of $Y$.
\end{example}
\begin{definition}
For a cosimplicial space $X^\bullet$, the \textit{totalization} is $Tot(X^\bullet) = \Hom_\Delta(\Delta^\bullet,X^\bullet) \subset \prod_{p=0}^\infty Map_{\cat{Top}} (\Delta^p,X^p)$. An element is a sequence $\{f_p:\Delta^p\ra X^p\}$ commuting with the coface and codegeneracy maps, i.e.
\begin{diagram}
\Delta^{p+1} & \rTo^{f_{p+1}} & X^{p+1} \\
\uTo^{d_i} & & \uTo_{d_i} \\
\Delta^p & \rTo_{f_p} & X^p \\
\dTo^{s_i} & & \dTo_{s_i} \\
\Delta^{p-1} & \rTo_{f_{p-1}} & X^{p-1}
\end{diagram}
commutes.
\end{definition}
This should be thought of as dual to geometric realization, which for a simplicial space $W_\bullet$ is given by $|W_\bullet|=W_\bullet \ox_\Delta \Delta^\bullet$. (In particular, totalization is a coend construction.) In fact, there is an adjunction relation between geometric realization and totalization.
\begin{example}
$Tot(\Delta^\bullet)$ is contractible.
\end{example}
\begin{example}
$Tot(\textup{cobar}(X)) \cong \Omega X$ and $Tot(\textup{cyclic cobar}(X))\cong LX$.
\end{example}
Now we can state our goal. \textbf{We would like to describe a structure that $(B^\bullet,X^\bullet)$ could have which induces an action of an $E_n$ operad (for $1\leq n\leq \infty$).} The plan for doing this is to start with $n=1$, move to $n=\infty$, and then interpolate between the two. As a bonus, we will also look at the framed little disks $f\mc{D}_2$.
\subsection{$n=1$}
Note that $N^*S^\bullet Y$ is a dga under the cup product, and there is a corresponding structure on $S^\bullet Y$. The cup product is given by $(x^p\smile y^q)(\sigma^{p+q}) = x(\sigma(0,\ldots, p)) \cdot y(\sigma(p,\ldots,p+q))$. Note that the $p$ is repeated; this will not always be the case in what follows. This gives $\smile:S^pY\otimes S^qY\ra S^{p+q}Y$. This satisfies the conditions:
\begin{enumerate}
\item \begin{equation*} d^i(x\smile y)= \left\{ \begin{array}{ll} d^ix\smile y, & i\leq p \\ x\smile d^{i-p}y, & i\ge p;\end{array} \right. \end{equation*}
\item $(d^{p+1}x)\smile y = x\smile (d^0y)$;
\item \begin{equation*} s^i(x\smile y) = \left\{ \begin{array}{ll} s^ix\smile y, & i\leq p-1 \\ x\smile s^{i-p}y, & i\geq p. \end{array} \right. \end{equation*}
\end{enumerate}
\begin{definition}
A \textit{cosimplicial abelian group/space with a cup product} is a cosimplicial abelian group/space with an associative and unital product satisfying the above properties.
\end{definition}
\begin{proposition}
The cup product on $B^\bullet$ induces a dga structure on $N^*(B^\bullet)$.
\end{proposition}
\begin{proof}
The proof is just checking that the cup product is a chain map.
\end{proof}
\begin{theorem}[\textbf{Theorem A}, Batanin, McClure-Smith]
The cup product on $X^\bullet$ induces an $A_\infty$ structure on $Tot(X^\bullet)$.
\end{theorem}
McClure considers this a \textit{useful theorem}, which he defines as a theorem with at least three applications. This gives, for example, that topological Hochschild cohomology has an $A_\infty$ structure (although in fact it has a $\mc{D}_2$ structure).
An interesting open problem is whether this is a Quillen equivalence. McClure conjectures that it does.
\noindent \textbf{Q.} Where's the $\mc{D}_2$ coming from in Deligne's conjecture?
\noindent \textbf{A.} A $\mc{D}_n$ action gives a Lie bracket on degree $n-1$, which is often called the \textit{Browder operation}. In the Hochschild complex we have a bracket of degree 1, which suggests that we want $n=2$. More generally, the primary obstruction to refining a $\mc{D}_n$ action to a $\mc{D}_{n+1}$ action is the Lie bracket. In fact, there are higher Hochschild cohomologies $HH^{(n)}(A)$ for $A$ an $E_n$-ring, and this has a $\mc{D}_{n+1}$ action.
\begin{exercise}
An \textit{operad with multiplication} is an operad $\mc{O}$ along with a map $As\ra\mc{O}$. For example, in the endomorphism operad of a ring $A$ we have $\mc{O}_A(p)=\Hom(A^{\ox p},A)$, and the map $As\ra \mc{O}_A$ is determined by $1\in \mc{O}_A(0)=A$ while the multiplication is determined by $M\in \mc{O}_A(2)=\Hom(A\ox A,A)$. Show that $1\in \mc{O}(0)=A$ induces $As\ra \mc{O}_A$, and show that an operad with multiplication induces a cosimplicial opject (with cup product).
\end{exercise}
\section{Generalizing Theorem A (which sits at $n=1$) to higher $n$}
One can continue along the path of $\smile_i$-products, but these end up being somewhat messy to work with, especially because of their signs. Rather, we will generalize in a different direction.
Recall that (up to signs), $(x\smile y)(\sigma) = x(\sigma(0,\ldots, p))\cdot y(\sigma(p,\ldots,p+q))$: the cup product has an interesting repetition. This leads us to the following reformulation of Theorem A.
For $x\in S^pY$, $y\in S^qY$, $\sigma\in S_{p+q+1}Y$, define $(x\sqcup y)(\sigma) = x(\sigma (0,\ldots, p))\cdot y(p+1,\ldots,p+q+1))$. This is not a chain map (and hence does not descend to cohomology), but it has obvious relations with the coface and codegeneracy maps. We call a \textit{cosimplicial abelian group/space with $\sqcup$} a cosimplicial space with operations as above that satisfy those same relations with the coface and coboundary maps.
\begin{proposition}
These two kinds of structure on a cosimplicial abelian group/space are equivalent.
\end{proposition}
Thus, there must be some $\sqcup$ in the Hochschild complex $C^\bullet A$; this is given by $(\varphi \sqcup \psi)(a_1\ox \cdots \ox a_{p+q+1}) = \varphi(a_1\ox \cdots \ox a_p)\cdot a_{p+1}\cdot \psi(a_{p+2}\ox \cdots \ox a_{p+1+q})$.
\subsection{The $E_\infty$ case}
Note that so far we have just been partitioning the vertices of $\sigma$; this is an associative sort of gadget. If we allow ourselves to mix the ordering, we will get a commutative sort of gadget.
We first need some notation.
\begin{definition}
The \textit{Delta category} $\Delta_+$ is the category of all finite totally-ordered sets, with morphisms bijections. (These will be denoted by the letters $S$ and $T$.) If $T\in \Delta_+$, we define the \textit{convex hull} of the elements in $T$ by $\Delta^T = \{h:t\ra \R:\sigma h(t) = 1, \ h(t)\geq 0 \ \forall t\} \subset Map_{\cat{Set}}(T,\R)$. We have the simplices $S_T Y= Map_{\cat{Top}}(\Delta^T,Y)$, and we have the cosimplices $S^TY=Map_{\cat{Set}}(S_T Y,\Z)$. (Note that $S^\emptyset Y \cong \Z$.)
\end{definition}
\begin{definition}
Given any $f:T\ra \{1,2\}$, we define $\langle f \rangle : S^{f^{-1}(1)}Y \otimes S^{f^{-1}(2)}Y \ra S^T Y$ by, for $\sigma:\Delta^T\ra Y$, setting $\brax{f}(x,y)(\sigma) = x(\sigma(f^{-1}(1)))\cdot y(\sigma(f^{-1}(2)))$.
\end{definition}
The idea is that $S^TY$ is essentially just $S^pY$, where $p=|T|-1$. This notation is also convenient because we don't need to put absolute values everywhere.
If $f:\{0,\ldots,p+q+1\}\ra \{1,2\}$ has $f^{-1}(1)=\{0,\ldots, p\}$ and $f^{-1}(2)=\{p+1,\ldots, p+q+1\}$ then we just recover $\brax{f}(x,y)=x\sqcup y$.
The family of $\brax{f}$ operations enjoys:
\begin{itemize}
\item a relation to the $d^i$ and $s^i$;
\item an appropriate sort of commutativity (given by permuting $\{1,2\}$);
\item an appropriate sort of associativity;
\item an appropriate unitality
\end{itemize}
The reason we must allow $\emptyset\in \Delta_+$ is to obtain the correct version of unitality. Otherwise, we would just be symmetric monoidal with no unit.
\begin{theorem}[Theorem B]
If $B^\bullet$ (or $X^\bullet$) has all $\brax{f}$ operations with the properties listed above, then this data induces an $E_\infty$ structure on $N^*B^\bullet$ (or $Tot(X^\bullet)$).
\end{theorem}
In particular, this gives us a new proof that the cochains on a topological space carry an $E_\infty$ structure.
\subsection{The $E_n$ case for $1p \end{array} \right. \\
s^i(x,y) &=& \left\{ \begin{array}{ll} (s^ix,y), & i