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\author{Gabriel C. Drummond-Cole}
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\title{Deformation Theory and Operads}
\begin{document}\maketitle
Recall that $D_k$ is the little $k$-disks operad.  It acts on $\Omega^k X$ for any topological space $X$.
\begin{thm}Weak Recognition principle (May 1972, Bordmann Vogt 1973)\\
If $X$ is a path connected topological space such that $X$ is an algebra over $D_k$, then there exists a space $Y$ so that $X\cong \Omega^k Y$, they are homotopy equivalent
\end{thm}
{\it Remark.}
\begin{enumerate}\item
There exists an operad $WD_k$ such that $X$ is an algebra over $WD_k$ if and only if $X$ is homotopy equivalent to a $k$-fold loop space $\Omega^k Y$.

$W$ is basically the free operad construction on $D_k$ in the topological category.
\item
$X$ is called an infinite loop space if there exists a sequence of spaces $X_1,X_2,\ldots$ such that $X\cong \Omega X_1$, $X_1\cong X_2$, and so on.  There is a strong recognition principle with the following.  A topological operad is called an $E_\infty$ operad if each $O(n)$ is $\Sigma_n$-free and each $O(n)$ is contractible.  Now $X$ is an algebra over an $E_\infty$ operad so that $X$ is an infinite loop space.  For example, take $D_\infty$ to be the direct limit of $D_k$, where $D_k$ sits inside $D_{k+1}$.  This limit is an $E_\infty$ operad and so you could, this answers your question.
\end{enumerate}

Let me give small indications for the weak recognition principle.  This was first done when $k=1$ by Stasheff.  Assume that $X$ is a $D_1$ space.  This is the little interval operad.  We want to show that $X$ is homotopy equivalent to $\Omega Y$.  This is in $H$-spaces in Lecture Notes in Mathematics in 1963.

This is in two steps.  The first step we show that there are polytopes $K_n$ for $n\ge 2$ and maps $K_n\to D_1(n)$.  This is the first example of this $W$.  The $K_n$ assemble to an operad $K$ where $K(n)=K_n\times \Sigma_n$ and $K$ is essentially equal to $WD_1$.  Spaces over this $K$ are called $A_\infty$ spaces.  Whenever you have an algebra over this $K$ you can define the following, let $Y$ be the union for $n\ge 0$ of $K_{n+2}\times X^n/\sim$, so you just, take this realization, put these together, and you divide out by equivalence and glue together and then one can check that $X$ is homotopy equivalent to $\Omega Y$.  Let me go back to step $1$.  The second step uses the notion of quasifibration and he builds this up from, starting up to $m$.  This uses a lot of topology machinery.  The nice thing that is really neat are these associahedra.  Define $K_n$ as the convex polytope with one vertex for each way of associating $n$ ordered variables $x_1,\ldots, x_n$.  For example, with six variables, I could associate with $x_1((x_2 x_3)(x_4(x_5 x_6)))$.  This is in correspondence with trivalent ordered trees.

What are the low dimensional examples?  $K_2$ is a point.  It's just, $(x_1 x_2)$.  Now $K_3$ is, well, you have two points, $(x_1 x_2)x_3$ and also $x_1(x_2 x_3)$ and you have to connect them and you get an edge.  $K_4$ there are five ways of associating
$$\xymatrix{
&&(x_1 x_2)(x_3 x_4)\ar@{-}[drr]\ar@{-}[dll]\\
x1(x2(x_3 x_4))\ar@{-}[dr]&&&&((x_1 x_2)x_3)x_4)\ar@{-}[dl]\\
&x_1((x_2 x_3)x_4)\ar@{-}[rr]&&(x_1(x_2x_3))x_4
}$$
Better than the convex polytope with this as its hull is to take $\sqcup K_r\times K_{n-r+1}/\sim$ and take the cone on this union $L_n$, I get something of dimension $n-2$.  One can check that $L_n$ has the homotopy type of the sphere, which is highly nontrivial.  This was shown by Adams.  I think I will stop here.

[Discussion of $\sim$]

[Picture of $K_5$]

Something I want to say is what happens when you take the homology of little disks.  Let $F:Top\to Vect$ be a functor respecting $(\times, \sqcup)\to (\otimes,\oplus)$.  If you have an operad in $\Cc$ then define $F(O)$ be the operad in $C'$ given by $F(O)=\{F(O(n))\}$ for $n\ge 1$.  You just drag everything through, so that $\circ_i=F(\circ_i)$ and $\Sigma_n$ action is just $F(\sigma_n)$, and the unit is $F(1_O)$.  I should have said that $*$ should map to $k$, it's a symmetric monoidal functor.  If it respects all the structure, then obviously all the relations are satisfied.  Then the proposition is
\begin{prop}
Let $k\ge 2$ and let $P_k=H_*(D_k)$.  Then a graded vector space $V$ is an algebra over $P_k$ if and only if $V$ is a $k-1$-algebra, where that is to be defined:
\end{prop}
\begin{defi}
To say that $V$ is a $(k-1)$-algebra means three things. 
\begin{enumerate}
\item $V$ is an associative and graded commutative algebra with product $\bullet$
\item $V[k-1]$ is a graded Lie algebra with bracket $\{\,,\,\}$ (the degree of the bracket is $1-k$)
\item There is a Liebnitz relation with the bracket $\{u,v\bullet w\}=\{u,v\}\bullet w +(-1)^{(|u|+k-1)|v|}v\bullet\{u,w\}$
\end{enumerate}
\end{defi}
What are these two operations?  Identify two homology elements.  Any old point as an element in $D_k(2)$, this is the product.  For the bracket, it's a $k-1$-sphere of spheres moving around a sphere.

There was one more thing I wanted to say, to clarify how you see an operad as a monoid in a monoidal category.  I'm not sure I should do that.  Let me do this, it's fairly fast.  It's an alternative definition of operads, of what an operad is.

Denote by $\Sigma-Vect$ the category with objects $A=\{A(n)\}_{n\ge 1}$ where each $A(n)$ has a $\Sigma_n$-right  action and morphisms are maps $f=\{f(n)\}_{n\ge 1}$ where $f_n:A(n)\to B(n)$ is an equivariant vector space map.  There is a monoidal structure called $\boxtimes$ on $\Sigma-Vect$ such that an operad on vector spaces is the same as a monoid in this monoidal category.

The idea is if you have, for $A$ and $B$ as having $n$ inputs, take $(A\boxtimes B) (n)$ to be trees with trees on them.  On the bottom you put $A$ and on the top $B$.  You take the sum over all trees with $n$ inputs, split into two levels with $B$s on top.  More precisely, you have to go through the following steps.  First of all, you extend $A(n)$ to any finite set $X$ as an input, so given $X$ and $A$ in $\Sigma-Vect$ let $A(X)$ be the sum over bijections $X\to n$ of $A(n)$ modulo some $\Sigma_n$ action.  Then in the second step, you can define for $n$ and $m$ at least $1$ and an object $A$, let $A(n,m)$ be the sum over surjective maps $n\to m$ of $A(f^{-1}(1))\otimes\cdots A(f^{-1}(m))$.  Finally, for the third step, you let $A\boxtimes B$ be the sum of $A(m)\otimes_{\Sigma_n} B(n,m)$.

The fact is that this $\Sigma-Vect, \boxtimes$ is a monoidal category with unit $U=U(1)=k$.  A monoid in $\Sigma-Vect$ is an object in $\Sigma-Vect$, an object $A$ together with morphisms $\mu:A\boxtimes A\to A$ and $\eta:U\to A$.  This should have the usual commutative diagrams for associativity of $\mu$ and the unit properties of $\eta$.  Such a monoid is exactly an operad.  Wednesday there is a seminar, Loday, so no class.


 
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