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\author{Gabriel C. Drummond-Cole}
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\title{Deformation Theory and Operads}
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Let $(A,D=\mu_1+\mu_2+\cdots)$ be an $A_\infty$ algebra with $\mu_1=:d_A$, and $(B,d_B)$ a chain complex.  Let $f:A\to B$ and $g:B\to A$ be chain maps, and $h:A\to A$ be a homotopy: $hd_A+d_Bh=gf-id_A$.  Then there exists 
\begin{itemize}
\item An $A_\infty$ structure $(B,\nu)$ extending $\nu_1=d_B$
\item an $A_\infty$ algebra map $F:TA[1]\to TB[1]$ with $F^1=f$.
\item an $A_\infty$ algebra map $G:TB[1]\to TA[1]$ with $G^1=g$.
\item an $A_\infty$ homotopy (I don't want to talk about this, really) $H:TA[1]\to TA[1]$ extending $h$ satisfying $HD+DH=GF-id_{TA[1]}$.
\end{itemize}
I used the following notation.  Did I use blue or red for $A$?
[Pictures]
You introduce three new terms when dragging.  You get terms with $fg$, terms with $h$ and a decomposition, and terms with the identity.
\begin{defi}
A Hodge decomposition of a chain complex $(V,d_V)$ is a decomposition $V=W\oplus B\oplus R$ such that $B=Im(d_V)$ and $W\oplus B=\ker (d_V)$. Hdet $H_.(V,d_V)=\frac{W\oplus B}{B}\cong W$ and $d_V(W\oplus B)=\{0\}$.  Then $d_V|_R:R\to B$ is an isomorphism since $B=Im d_V$ and $R\cap W\oplus B=\{0\}$.
\end{defi}
Note also that $f:W\oplus B\oplus R\to W$ and the reverse inclusion are homotopy inverses: $fg-id_W$ is $0$  and $gf-id_V$ is the projection onto $B\oplus R$.  It's not hard to see that this is $-dd^{-1}-d^{-1}d$.  Then every finite dimensional chain complex has a Hodge decomposition.
\begin{cor}
Kadeishvili 82 (The algebra structure on the homology of an $A_\infty$ algebra\\
Let $(A,D)$ be an $A_\infty$ algebra and let $H$ be the homology of $A$ with respect to $\mu_1$.  Then there exists an $A_\infty$ structure on $H$ with $\nu_1=0$ and $A_\infty$ algebra maps $F:TA[1]\to TH[1]$ and $G:TH[1]\to TA[1]$ and a homotopy $H$ with $GF-id=HD+DH$. \end{cor}
This comes from choosIng a Hodge decomposition and applying the transfer theorem.  For example, $\nu_n$ is the sum over all kinds of trees where you put things from $H$ to $V$, use products in $V$, intersperse with homologies $d^{-1}$, and eventually project back.


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