rms can be put together as before to produce a map $\chi_i$, so that now $\Upsilon_{i+1}:=\chi_2+\dots+\chi_i$ only maps into $M_{i+1}\oplus M_{i+2}\oplus\dots $. We therefore obtain the chain map $\chi$, and with this, we define the homotopy $\mathcal Comm$-inner product as $f:=\chi(\mu)\in  Mod(F_{\mathcal Lie, C[1]}C^*[1],F_{\mathcal Lie, C[1]}C[1]) $, where $\mu\in C$ denotes the fundamental cycle of the space $X$. Since $\mu$ is $d_1$-closed, it follows that $f\circ h-g\circ f=0$.

The operad $\widehat{\mathcal O}$ {#cyclic-op}
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In this section, we define for any cyclic operad $\mathcal O$ the colored operad $\widehat{\mathcal O}$. In the case that $\mathcal O$ is cyclic quadratic, we give an explicit description of $\widehat{\mathcal O}$ in terms of generators and relations coming from generators and relations in $\mathcal O$.

We assume that the reader is familiar with the notion of operads, colored operads and cyclic operads. For a good introduction to operads, we refer to [@Ad], [@GK] and [@MSS], for cyclic operads we recommend [@GeK] and [@MSS]. Colored operads were first introduced in [@BV] and appeared in many other places, see e.g. [@L] and [@BM]. Since in our case, we only need a special type of colored operad, it will be convenient to setup notation with the following definition.

As in [@GK (1.2.1)] and [@GeK (1.1)], we assume throughout this paper that $k$ is a field of characteristic $0$. Note however, that for certain operads such as e.g. the associative operad, a more general setup is possible.

\[0/1-operad\] Let $\mathcal P$ be a 3-colored operad in the category of (differential graded) vector spaces, where we use the three colors “full", “dashed" and “empty", in symbols written ${
 \begin{pspicture}(0,0.1)(0.2,0.4)
 \psline[linewidth=1pt](0.1,0)(0.1,0.4)
 \end{pspicture}},{
 \begin{pspicture}(0,0.1)(0.2,0.4)
  \psline[linewidth=1pt, linestyle=dashed,dash=4pt 3pt](0.1,0)(0.1,0.4)
 \end{pspicture}},\varnothing$. This means that to each finite sequence o