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\TopicFR{GÃ©omÃ©trie algÃ©brique, GÃ©omÃ©trie et Topologie}
\TopicEN{Algebraic geometry, Geometry and Topology}

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\title{The (co)homology of the Deligne--Mumford moduli spaces of marked rational curves}
\alttitle{La (co)homologie des espaces de moduli de Deligne--Mumford de courbes rationnelles marquÃ©es}

\author{\firstname{Aleksey} \lastname{Zinger}}
\address{Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, United States of America}
\email{azinger@math.stonybrook.edu}

\begin{abstract}
This informal note collects key results and open problems on the (co)homology of the Deligne--Mumford moduli spaces of real marked rational curves. The open problems are both of topological nature, aiming to investigate the (co)homology of these spaces further, and of algebraic nature, aiming to describe the structure of the homology of these spaces in operad-like terms.
\end{abstract}

\begin{altabstract}
Cette note informelle rassemble des rÃ©sultats clÃ©s et des problÃ¨mes ouverts sur la (co)homologie des espaces de moduli de Deligne--Mumford des courbes rationnelles marquÃ©es rÃ©elles. Les problÃ¨mes ouverts sont Ã&nbsp; la fois de nature topologique, visant Ã&nbsp; approfondir l'Ã©tude de la (co)homologie de ces espaces, et de nature algÃ©brique, visant Ã&nbsp; dÃ©crire la structure de l'homologie de ces espaces en termes d'opÃ©rades.
\end{altabstract}

\begin{document}

\maketitle

This note includes a readily accessible presentation of the recent results of~\cite{RDMhomol,RDMbl} on the topology of real Deligne--Mumford moduli spaces. These results are the main outcome of a decade-old aim, the realization of which also includes the systematic treatment of blowups in~\cite{RDMbl2}, and are the main topological inputs in~\cite{RGWsAlg}. However, the primary motivation for this note is an algebraic reformulation of the results of~\cite{RDMhomol,RDMbl}, in terms of operads, which was inspired by~\cite{EHKR} and the analogous reformulation of the main results of~\cite{Keel} in~\cite{Getz95}. We also include open problems, of both purely topological nature and purely algebraic, with the latter based on the operad formalism.

Below are the two moduli spaces central to this note.

$\ovcM_{\ell}=$ Deligne--Mumford moduli space of (complex) rational curves with $\ell$~marked points; smooth projective variety of complex dimension~$\ell-3$. An element of this space is the equivalence class of stable tuples
\[
\cC\equiv\parens{\Si,z_1,\dots,z_{\ell}},
\]
where $\Si$ is a connected, possibly nodal, Riemann surface and $z_1,\dots,z_{\ell}\in \Si$ are distinct smooth points.

$\R\ovcM_{k,\ell}=$ Deligne--Mumford moduli space of real rational curves with $k$~real marked points and $\ell$ conjugate pairs of marked points; compact smooth real analytic variety of dimension~$k+2\ell-3$; orientable if $k = 0$ or $k+2\ell \le 4$. An element of this space is the equivalence class of stable tuples
\[
\cC\equiv\parens[\big]{\Si,\si,z_1,\dots,z_k,(z_1^+,z_1^-),\dots,(z_{\ell}^+,z_{\ell}^-)},
\]
where $\Si$ is a connected, possibly nodal, Riemann surface, $\si$ is an anti-holomorphic involution on~$\Si$ and
\[
z_1,\dots,z_k,z_1^+,z_1^-,\dots,z_{\ell}^+,z_{\ell}^-\in\Si
\]
are distinct smooth points such that $\si(z_i^+) = z_i^-$ for $i = 1,\dots,\ell$.

For $\ell\in \Z^{\ge0}$, we define $[\ell] = \{1,\dots,\ell\}$.

\section{Key results}

\begin{proof}[Complex case, $\ovcM_{\ell}$]
This smooth projective variety contains smooth divisors~$D_{J,K}$, with $J \sqcup K = [\ell]$ and $\abs{J},\abs{K} \ge 2$, whose generic element is a two-component curve with the marked points on the two components indexed by~$J$ and~$K$; see Figure~\ref{M04rel_fig} for some examples. It is a now classical result of Keel~\cite{Keel} that
\begin{enumerate}\renewcommand{\theenumi}{\ensuremath{\C}\arabic{enumi}}\relabel
\item\label{Cgen_it} these divisors generate $H^*(\ovcM_{\ell};\Z)$ as an algebra,
\item\label{Crel_it} subject only to some obvious relations;
\end{enumerate}
see~\cite[Theorem~2.1]{RDMhomol} for a precise statement. The relations arise from
\begin{enumerate}\renewcommand{\theenumi}{\ref*{Crel_it}\alph{enumi}}\relabel
\item certain divisors~$D_{J,K} \subset \ovcM_{\ell}$ being disjoint and
\item all divisors~$D_{J,K}$ in~$\ovcM_4 \approx \C\P^1$ being points and thus equivalent in $H^2(\ovcM_4;\Z)$, as indicated in Figure~\ref{M04rel_fig}.
\end{enumerate}
The above relations in $H^2(\ovcM_4;\Z)$ pull back to relations in $H^2(\ovcM_{\ell};\Z)$ with $\ell \ge 4$ via the forgetful morphisms $\ovcM_{\ell} \to \ovcM_4$ dropping all but four of the marked points (every 4-element subset of~$[\ell]$ determines such a morphism).

\begin{figure}[h]
\includegraphics[width=.6\linewidth]{M04rel_fig}
\caption{Relations between the divisors $D_{J,K} \subset \ovcMcapt_4$ in $H^2(\ovcMcapt_4;\Z)$.\label{M04rel_fig}}
\end{figure}

The proof in~\cite{Keel} presents $\ovcM_{\ell+1}$ as a sequence of holomorphic blowups of~$\ovcM_{\ell} \times \ovcM_4$ and determines $H^*(X_{\vr};\Z)$ inductively for every intermediate blowup~$X_{\vr}$, assuming the claimed result for~$H^*(\ovcM_{\ell};\Z)$. The proof of the main result of~\cite{Keel} given in~\cite{RDMhomol} still uses this sequence of blowups to obtain~\eqref{Cgen_it}, but establishes~\eqref{Crel_it} directly, without determining $H^*(X_{\vr};\Z)$ for the intermediate blowups~$X_{\vr}$.

For any commutative ring $R$ with unity~1, the modules $H_*(\ovcM_{\ell};R)$ can be arranged into an operad of graded $R$-modules
\[
\cO_{\C}(\ell)\equiv
\begin{cases}
R,						&amp; \text{if} \ \ell = 1;\\
H_*(\ovcM_{\ell+1};R),	&amp; \text{if} \ \ell \ge 2;
\end{cases}
\]
as follows. For integers $k,\ell \ge 2$ and $i\in [k]$, let
\[
\io_{k\ell i} \colon \ovcM_{k+1} \times \ovcM_{\ell+1} \to \ovcM_{k+\ell}
\]
be the immersion identifying the $i$-th marked point of a marked curve $[\cC_1]\in \ovcM_{k+1}$ with the last marked point of a marked curve $[\cC_2]\in \ovcM_{\ell+1}$, keeping the indices of the first $i-1$ marked points of~$[\cC_1]$ the same, increasing the indices of the last $k+1-i$ marked points of~$[\cC_1]$ by~$\ell-1$, and increasing the indices of the first $\ell$~marked points of~$[\cC_2]$ by~$i-1$. This induces homomorphisms
\[
\io_{k\ell i*} \colon \cO_{\C}(k) \times \cO_{\C}(\ell) \approx H_*\parens[\big]{\ovcM_{k+1} \times \ovcM_{\ell+1}; R} \to \cO_{\C}(k + \ell - 1).
\]
For $k,\ell\in \Z^+$ and $i\in [k]$, define
\begin{equation}\label{prodcases_e}
\begin{aligned}
\io_{k1i*} \colon \cO_{\C}(k) \times \cO_{\C}(1) &amp; \to \cO_{\C}(k), 			\quad 	&amp;&amp; \io_{k1i*}(\eta,r) = r\eta,\\
\io_{1\ell 1*} \colon \cO_{\C}(1) \times \cO_{\C}(\ell) &amp; \to \cO_{\C}(\ell),  		&amp;&amp; \io_{1\ell 1*}(r,\eta) = r\eta.
\end{aligned}
\end{equation}
For $k,\ell_1,\dots,\ell_k\in \Z^+$, define
\begin{equation}\label{expandcomp_e}
\begin{gathered}
\circ \colon \cO_{\C}(k) \times \cO_{\C}(\ell_1) \times \dots \times \cO_{\C}(\ell_k) \to \cO_{\C}\parens{\ell_1 + \dots + \ell_k},\\
\eta \circ (\eta_1,\dots,\eta_k) =\io_{(1+\ell_2+\dots+\ell_k)\ell_11*}
\parens[\bigg]{\cdots\io_{(k-1+\ell_k)\ell_{k-1}(k-1)*}\parens[\big]{\io_{k\ell_k k*}(\eta,\eta_k),\eta_{k-1}},
\dots,\eta_1}.
\end{gathered}
\end{equation}
The resulting operad $\parens[\big]{\parens{\cO_{\C}(\ell)}_{\ell\in\Z^+},\circ}$ is the operad of hypercommutative algebras; see~\cite{Getz95}.
\let\qed\relax
\end{proof}

\begin{proof}[Real case with real points only, $\R\ovcM_{k,0}$]
By the main result of~\cite{Krasnov}, $H^*(\R\ovcM_{k,0};\Z_2)$ is described in the same way as $H^{2*}(\ovcM_k;\Z)$ above with the complex subvarieties $D_{J,K} \subset \ovcM_k$ replaced by the analogous real subvarieties
$\R D_{J,K} \subset \R\ovcM_{0,k}$; see the proof of~\cite[Theorem~5.6]{EHKR}. Since $\R\ovcM_{k,0}$ is not orientable for $k \ge 5$, the homology and cohomology of~$\R\ovcM_{k,0}$ with coefficients in a ring of characteristic larger than~2 are quite distinct. For any commutative ring~$R$ with unity and distinct $a,b,c,d\in [k]$, $H^1(\R\ovcM_{k,0};R)$ contains the pullback~$\om_{abcd}$ of the PoincarÃ© dual of a point in $\R\ovcM_{4,0} \approx \R\P^1$ by the forgetful morphism
$\R\ovcM_{k,0} \to \R\ovcM_{4,0}$ sending the marked points indexed by $a$, $b$, $c$, $d$ to the marked points indexed by~$1$, $2$, $3$, $4$, respectively, and dropping the remaining marked points. This class is PoincarÃ© dual to a linear combination of the real hypersurfaces~$\R D_{J,K}$.

By~\cite[Theorem~2.9]{EHKR},
\begin{enumerate}\renewcommand{\theenumi}{\ensuremath{\R\R}\arabic{enumi}}\relabel
\item\label{RRgen_it} the classes~$\om_{abcd}$ generate $H^*(\R\ovcM_{k,0};\Q)$ as an algebra,
\item\label{RRrel_it} subject only to some expected relations.
\end{enumerate}
The relations arise from
\begin{enumerate}\renewcommand{\theenumi}{\ref*{RRrel_it}\alph{enumi}}\relabel
\item the diffeomorphism of $\R\ovcM_{4,0}$ induced by the interchange of two marked points being orientation-reversing and
\item $\R\ovcM_{5,0}$ being diffeomorphic to the blowup of $\R\P^1 \times \R\P^1$ at three points of the diagonal.
\end{enumerate}
The resulting relations in $H^1(\R\ovcM_{4,0};\Q)$, $H^1(\R\ovcM_{5,0};\Q)$, and $H^2(\R\ovcM_{5,0};\Q)$ pull back to relations in $H^1(\R\ovcM_{k,0};\Q)$ and $H^2(\R\ovcM_{k,0};\Q)$ via the forgetful morphisms
\[
\R\ovcM_{k,0} \to \R\ovcM_{4,0} \qquad\text{and}\qquad \R\ovcM_{k,0} \to \R\ovcM_{5,0}
\]
dropping all but four (resp.~five) of the marked points. By~\cite[Corollary~6.9]{EHKR}, certain natural subvarieties of~$\R\ovcM_{k,0}$ form a linear basis for~$H_*(\R\ovcM_{k,0};\Q)$. By~\cite[Corollary~3.8]{Rains}, $H_*(\R\ovcM_{k,0};\Z)$ and $H^*(\R\ovcM_{k,0};\Z)$ have only 2-torsion; it is described in~\cite[Section~5]{EHKR}.

Similarly to the complex case, there are node-identifying immersions
\[
\io_{k\ell i}^{\R} \colon \R\ovcM_{k+1,0} \times \R\ovcM_{\ell+1,0} \to \R\ovcM_{k+\ell,0}.
\]
They again determine an operad structure on the sequence
\[
\cO_{\R\R}(k)\equiv
\begin{cases}
\Q,							&amp; \text{if} \ k = 1;\\
H_*(\R\ovcM_{k+1,0};\Q),		&amp; \text{if} \ k \ge 2;
\end{cases}
\]
of graded vector spaces. By~\cite[Theorem~2.14]{EHKR}, this operad is the operad of unital 2-Gerstenhaber algebras.

The proof of~\cite[Corollary~3.8]{Rains} is based on presenting~$\R\ovcM_{k,0}$ as a sequence of real blowups of~$(\R\P^1)^{k-3}$ and shows that $H_*(X_{\vr};\Z)$ has only 2-torsion for every intermediate blowup~$X_{\vr}$. Theorems~2.9 and~2.12 and Corollary~6.9 in~\cite{EHKR} are proved in parallel, by combining the aforementioned topological results of~\cite{Keel,Krasnov} with algebraic considerations. The same results should be obtainable by presenting~$\R\ovcM_{k+1,0}$ as a sequence of real blowups of~$\R\ovcM_{k,0} \times \R\ovcM_{4,0}$ as in~\cite{RDMbl} and relating the (co)homologies of the intermediate blowups as in~\cite[Section~3.2]{RDMhomol}.
\let\qed\relax
\end{proof}

\begin{proof}[Real case with conjugate points only, $\R\ovcM_{0,\ell}$]
This orientable compact smooth manifold contains compact real hypersurfaces~$\R E_{J,K}$, with $J \sqcup K = [\ell]$, and~$\R H_{J,K}$, with $J \sqcup K = [\ell]$ and $J,K \neq \eset$, whose generic elements are curves with two irreducible components that are interchanged by the involution in the first case and are preserved by~it in the second case. This manifold also contains compact submanifolds~$\R D_{I;J,K}$, with $I \sqcup J \sqcup K = [\ell]$, $I \neq \eset$, and $\abs{J}+\abs{K} \ge 2$, of (real) codimension~2 whose generic elements are curves with three components two of which are interchanged by the involution; see Figure~\ref{TopolRel_fig} for some examples. The submanifolds~$\R E_{J,K}$ and~$\R D_{I;J,K}$ are orientable, while~$\R H_{J,K}$ is not if $k \ge 3$.

\begin{figure}[h]
\includegraphics[width=.9\linewidth]{TopolRel_fig}
\caption{The first line represents an equivalence of two points in $H^1(\R\ovcMcapt_{0,2};\Q)$ and a relation between three loops in $H^2(\R\ovcMcapt_{0,3};\Q)$. The bottom diagram represents the intersection pattern of the six codimension~2 submanifolds $\R D_{I;J,K} \approx S^1$ with the four real hypersurfaces $\R E_{J',K'} \approx S^2$ in~$\R\overline\cM_{0,3}$. The former are labeled by the unique element of $I \subset [3]$ and the sign of $(-1)^{\abs{J}} = (-1)^{\abs{K}}$; the latter are labeled by the subset $J',K' \subset [3]$ not containing~1.\label{TopolRel_fig}}
\end{figure}

By~\cite[Theorem~2.2]{RDMhomol},
\begin{enumerate}\renewcommand{\theenumi}{\ensuremath{\R\C}\arabic{enumi}}\relabel
\item\label{RCgen_it} the submanifolds~$\R E_{J,K}$ and~$\R D_{I;J,K}$ generate $H^*(\R\ovcM_{0,\ell};\Q)$ as an algebra,
\item\label{RCrel_it} subject only to some expected relations.
\end{enumerate}
The relations arise from
\begin{enumerate}\renewcommand{\theenumi}{\ref*{RCrel_it}\alph{enumi}}\relabel
\item certain submanifolds $\R E_{J,K},\R D_{I;J,K} \subset \R\ovcM_{0,\ell}$ being disjoint,
\item all real hypersurfaces~$\R E_{J,K}$ in~$\R\ovcM_{0,2} \approx \R\P^1$ being points and thus equivalent in $H^1(\R\ovcM_{0,2};\Q)$, as indicated by the left identity in Figure~\ref{TopolRel_fig}, and
\item the relation in $H_1(\R\ovcM_{0,3};\Q)$, or equivalently in $H^2(\R\ovcM_{0,3};\Q)$, of Remark~3.4 in~\cite{RealEnum}, as indicated by the right identity and the bottom diagram in Figure~\ref{TopolRel_fig}.
\end{enumerate}
The above relations in $H^1(\R\ovcM_{0,2};\Q)$ and $H^2(\R\ovcM_{0,3};\Q)$ pull back to relations in $H^1(\R\ovcM_{0,\ell};\Q)$ and $H^2(\R\ovcM_{0,3};\Q)$ via the forgetful morphisms
\[
\R\ovcM_{0,\ell} \to \R\ovcM_{0,2} \qquad\text{and}\qquad \R\ovcM_{0,\ell} \to \R\ovcM_{0,3}
\]
dropping all but two (resp.~three) conjugate pairs of marked points.

The proof of~\eqref{RCgen_it} in~\cite{RDMhomol} is based on the presentation of $\R\ovcM_{0,\ell+1}$ in~\cite{RDMbl} as a sequence of blowups of~$\R\ovcM_{0,\ell} \times \ovcM_4$ of three different types and describes generators for~$H^*(X_{\vr};\Q)$ for every intermediate blowup~$X_{\vr}$, assuming~\eqref{RCgen_it} for~$\R\ovcM_{0,\ell}$. This approach can also be used to describe generators for~$H_*(\R\ovcM_{0,\ell};\Z)$ and implies that this module has only 2-torsion. The proof of~\eqref{RCrel_it} in~\cite{RDMhomol} relates $H^*(\R\ovcM_{0,\ell+1};\Q)$ to $H^*(\R\ovcM_{0,\ell'};\Q)$ with $\ell' \le \ell$ and $H^*(\ovcM_{\ell'};\Q)$ with $\ell' \le \ell+1$ by pairing certain cohomology classes with the submanifolds~$\R E_{J,K}$ and~$\R D_{I;J,K}$ to show the algebraic independence of the former.

Let $R$ be a commutative ring with unity~1 and
\[
\cO_{\R\C}(k)=H_*\parens{\R\ovcM_{0,k+1};R} \quad \forall k\in \Z^+.
\]
For integers $k \ge 1$, $\ell \ge 2$, and $i\in [k]$, let
\[
\io_{k\ell i}^+ \colon \R\ovcM_{0,k+1} \times \ovcM_{\ell+1} \to \R\ovcM_{0,k+\ell}
\]
be the immersion identifying the first marked point in the $i$-th conjugate pair of a marked curve $[\cC_1]\in \R\ovcM_{0,k+1}$ with the last marked point of a marked curve $[\cC_2]\in \ovcM_{\ell+1}$, and the second marked point in this pair with the last marked point of the conjugate marked curve~$[\ov{\cC_2}]$, keeping the indices of the first $i-1$ conjugate pairs of marked points of~$[\cC_1]$ the same, increasing the indices of the last $k+1-i$ conjugate pairs of marked points of~$[\cC_1]$ by~$\ell-1$, and turning the first $\ell$~marked points of~$[\cC_2]$ into the first marked points of the conjugate pairs with the indices increased by~$i-1$; see Figure~\ref{ioplus_fig} for an example. This induces homomorphisms
\[
\io_{k\ell i*}^+ \colon \cO_{\R\C}(k) \times \cO_{\C}(\ell) \approx H_*\parens[\big]{\R\ovcM_{0,k+1} \times \ovcM_{\ell+1};R} \to \cO_{\R\C}(k+\ell-1).
\]
For $k\in \Z^+$ and $i\in [k]$, define
\[
\io_{k1i*}^+ \colon \cO_{\R\C}(k) \times \cO_{\C}(1) \to \cO_{\R\C}(k),
\qquad \io_{k1i*}^+(\eta,r)=r\eta.
\]
For $k,\ell_1,\dots,\ell_k\in \Z^+$, define
\[
\begin{gathered}
\circ \colon \cO_{\R\C}(k) \times \cO_{\C}(\ell_1) \times \dots \times \cO_{\C}(\ell_k) \to \cO_{\R\C}(\ell_1 + \dots + \ell_k),\\
\eta \circ (\eta_1,\dots,\eta_k) =\io_{(1+\ell_2+\dots+\ell_k)\ell_11*}^+ \parens[\bigg]{\cdots\io_{(k-1+\ell_k)\ell_{k-1}(k-1)*}^+\parens[\big]{\io_{k\ell_kk*}^+(\eta,\eta_k),\eta_{k-1}},\dots,\eta_1}.
\end{gathered}
\]
This determines a right module structure on $\parens{\cO_{\R\C}(k)}_{k\in\Z^+}$ over the operad $\parens[\big]{(\cO_{\C}(\ell)}_{\ell\in\Z^+},\circ)$.
\let\qed\relax
\end{proof}

\begin{figure}[h]
\includegraphics[width=.7\linewidth]{ioplus_fig}
\caption{The action of the immersion~$\io_{232}^+$ on a typical element in its domain; each double-headed arrow labeled~$\si$ indicates the involution on the corresponding real curve.\label{ioplus_fig}}
\end{figure}

\section{Open problems}

\begin{proof}[Real case with conjugate points only, $\R\ovcM_{0,\ell}$]
The approach of~\cite{RDMbl,RDMhomol} to~\eqref{RCgen_it} implies that $H^*(\R\ovcM_{0,\ell};\Z)$ is linearly generated by orientable intersections of the submanifolds $\R E_{J,K}$, $\R H_{J,K}$, and~$\R D_{I;J,K}$ and contains only 2-torsion. For example, the torsion of $H_1(\R\ovcM_{0,3};\Z)$ is generated by $\R H_{\{1,2\},\{3\}} \cap \R H_{\{1\},\{2,3\}}$, as shown in~\cite[Section~3]{RealEnum}. However, it still remains to determine the amount of 2-torsion and the ring structure of~$H^*(\R\ovcM_{0,\ell};\Z)$ for $\ell \ge 4$. The algebraic generators for the torsion of~$H_*(\R\ovcM_{0,\ell};\Z)$ should correspond to trees, with the univalent and bivalent vertices labeled by nonempty subsets of~$[\ell]$ forming a partition of~$[\ell]$. A more algebraic question is to describe the structure of the right module structure~$\parens{\cO_{\R\C}(k)}_{k\in\Z^+}$ over the operad $\parens[\big]{(\cO_{\C}(\ell))_{\ell\in\Z^+},\circ}$ explicitly, in the spirit of~\cite[Theorem~2.14]{EHKR} and~\cite{Getz95}.
\let\qed\relax
\end{proof}

\begin{proof}[The general real case, $\R\ovcM_{k,\ell}$ with $k,\ell\in \Z^+$]
The approach of~\cite{RDMbl,RDMhomol} to~\eqref{RCgen_it} also implies that $H_*(\R\ovcM_{k,\ell};\Z)$ with $k \ge 1$ is linearly generated by orientable intersections of the submanifolds $\R H_{J,K}$ and~$\R D_{I;J,K}$ and contains only 2-torsion. The former are now indexed by pairs $J \equiv (J_{\R},J_{\C})$ and $K \equiv (K_{\R},K_{\C})$~with
\[
[k] = J_{\R} \sqcup K_{\R},
\qquad [\ell] = J_{\C} \sqcup K_{\C},
\qquad \text{and}
\qquad \abs{J_{\R}} + 2\abs{J_{\C}}, \ \abs{K_{\R}} + 2\abs{K_{\C}}\ge 2,
\]
and the latter by the partitions $[\ell] = I \sqcup J \sqcup K$ with $\abs{J}+\abs{K} \ge 2$. Similarly, $H^*(\R\ovcM_{k,\ell};\Z)$ is generated by some linear combinations of these intersections. In fact, the most delicate type of blowups appearing in~\cite{RDMbl,RDMhomol} no longer enters into consideration. However, it remains to determine the amount of 2-torsion in~$H_*(\R\ovcM_{k,\ell};\Z)$ and the ring structure even of~$H^*(\R\ovcM_{k,\ell};\Q)$.

For $k\in \Z^+$ and $\ell\in \Z^{\ge0}$, let
\[
\cO_{\R}(k,\ell) =
\begin{cases}
\Q,								&amp; \text{if} \ k+\ell = 1;\\
H_*(\R\ovcM_{k+1,\ell};\Q),		&amp; \text{if} \ k+\ell \ge 2;
\end{cases}
\qquad\qquad \cO_{\R}(k)=\bigoplus_{\ell=0}^{\infty} \cO_{\R}(k,\ell).
\]
Similarly to the previous cases, there are node-identifying immersions
\[
\io_{k\ell k'\ell' i}^{\R} \colon \R\ovcM_{k+1,\ell} \times \R\ovcM_{k'+1,\ell'} \to \R\ovcM_{k+k',\ell+\ell'}
\]
and
\[
\io_{k\ell\ell' i}^+ \colon \R\ovcM_{k+1,\ell} \times \ovcM_{\ell'+1} \to \R\ovcM_{k+1,\ell+\ell'-1},
\]
with $k+\ell \ge 2$ in both cases, $i\in [k+1]$ and $k'+\ell' \ge 2$ in the first case, and $i\in [\ell]$ and $\ell' \ge 2$ in the last case. They induce homomorphisms
\[
\io_{k\ell k'\ell' i*}^{\R} \colon \cO_{\R}(k,\ell) \times \cO_{\R}(k',\ell') \approx  H_*\parens[\big]{\R\ovcM_{k+1,\ell} \times \R\ovcM_{k'+1,\ell'};\Q} \to \cO_{\R}(k+k'-1,\ell+\ell')
\]
and
\[
\io_{k\ell\ell' i*}^+ \colon \cO_{\R}(k,\ell) \times \cO_{\C}(\ell') \approx  H_*\parens[\big]{\R\ovcM_{k+1,\ell} \times \ovcM_{\ell'+1};\Q} \to \cO_{\R}(k,\ell+\ell'-1).
\]
We define the homomorphisms
\begin{align*}
&amp; \io_{k\ell10i*}^{\R} \colon \cO_{\R}(k,\ell) \times \cO_{\R}(1,0) \to \cO_{\C}(k,\ell),\\
&amp; \io_{10k'\ell' 1*}^{\R} \colon \cO_{\R}(1,0) \times \cO_{\R}(k',\ell') \to \cO_{\C}(k',\ell'),\\
\text{and}\quad
&amp; \io_{k\ell 1 i*}^+ \colon \cO_{\R}(k,\ell) \times \cO_{\C}(1) \to \cO_{\C}(k,\ell)
\end{align*}
similarly to~\eref{prodcases_e}. These bilinear maps induce multilinear maps
\[
\circ \colon \cO_{\R}(k,\ell) \times \cO_{\R}(m_1,\ell_1') \times \dots \times \cO_{\R}(m_k,\ell_k') \to \cO_{\R}(m_1 + \dots + m_k,\ell + \ell_1' + \dots + \ell_k')
\]
and
\[
\circ \colon \cO_{\R}(k,\ell) \times \cO_{\C}(m_1) \times \dots \times \cO_{\C}(m_{\ell}) \to 
\cO_{\R}(k,m_1 + \dots + m_{\ell})
\]
similarly to~\eref{expandcomp_e}. The last two maps determine an operad $\parens{\cO_{\R}(k)}_{k\in\Z^+}$ of graded vector spaces and a right module structure on $\parens{\cO_{\R}(k,\ell)}_{\ell\in\Z^+}$ over the operad $\parens[\big]{\parens{\cO_{\C}(\ell)}_{\ell\in\Z^+},\circ}$ for each $k\in \Z^+$.

As suggested by A.~Voronov, the above structures can be combined into an object resembling a bicolored operad; see~\cite[Section~1.1]{Gir}. For $n\in \Z^+$, let
\[
\cO_{\R}'(n) = \cO_{\C}(n) \sqcup \bigsqcup_{\substack{k+\ell=n \\ k\ge0, \ \ell\ge1}} \cO_{\R}(k+1,\ell-1).
\]
For each fixed $n\in \Z^+$, auxiliary grading is provided by $k\in \Z^{\ge0}$ above, with the elements of~$\cO_{\C}(n)$ being of the auxiliary degree~0. For $x\in \cO_{\R}'(n)$, let $\abs{x} = n$ and
\[
\abs{x}_+=
\begin{cases}
n,			&amp; \text{if} \ x\in \cO_{\C}(n);\\
\ell-1,		&amp; \text{if} \ x\in \cO_{\R}(k+1,\ell-1).
\end{cases}
\]
We take the set of colors to be $\cC \equiv \{+,\R\}$ and define
\[
\opout \colon \cO_{\R}'(n) \to \cC,
\qquad \opout(x) =
\begin{cases}
+,		&amp; \text{if} \ x\in \cO_{\C}(n);\\
\R,		&amp; \text{if} \ x\in \cO_{\R}(k+1,\ell-1).
\end{cases}
\]
For $i\in [n]$, define
\[
\opini \colon \cO_{\R}'(n) \to \cC,
\qquad \opini(x) =
\begin{cases}
+,		&amp; \text{if} \ i \le \abs{x}_+;\\
\R,		&amp; \text{if} \ i &gt; \abs{x}_+.
\end{cases}
\]
Let $1_+ = 1\in \cO_{\C}(1)$ and $1_{\R} = 1\in \cO_{\R}(1,0)$. The above auxiliary grading is preserved by the~maps
\[
\circ_i \colon \braces[\big]{(x,y)\in \cO_{\R}'(m) \times \cO_{\R}'(n) \;:\; \opini(x) = \opout(y)} \to \cO_{\R}'(m+n-i),
\qquad i\in [m],
\]
induced by the homomorphisms~$\io_{k\ell k'\ell' i*}^{\R}$ and $\io_{k\ell\ell' i*}^+$ defined in the previous paragraph. These maps satisfy a modification of~\cite[(1.1.4a)]{Gir},
\begin{equation}\label{114a_e}
x \circ_i (y \circ_j z) =
\begin{cases}
(x \circ_i y) \circ_{j+i-1} z,			&amp; \text{if} \ \opout(y) = \opout(z);\\
(x \circ_i y) \circ_{j+\abs{x}_+} z,		&amp; \text{if} \ \opout(y) \neq \opout(z);
\end{cases}
\qquad\qquad \text{if} \ 1\le i\le \abs{x},\ 1\le j\le \abs{y}.
\end{equation}
These maps also satisfy a modification of~\cite[(1.1.4b)]{Gir},
\begin{equation}\label{114b_e}
(x \circ_i y) \circ_{j+\abs{y}-1} z =
\begin{cases}
(x \circ_j z) \circ_i y,				&amp; \text{if} \ \opout(y) = +;\\
(x \circ_j z) \circ_{i+\abs{z}_+} y,	&amp; \text{if} \ \opout(y) = \R;
\end{cases}
\qquad\qquad \text{if} \ 1\le i&lt; j\le \abs{x}.
\end{equation}
The identities~\eref{114a_e} and~\eref{114b_e} hold whenever both sides are defined. They agree with~\cite[(1.1.4a)]{Gir} and~\cite[(1.1.4b)]{Gir} whenever~$\opout(y) = \opout(x)$ and $\opout(y) = \R$, respectively. The identity~\cite[(1.1.4c)]{Gir} holds without any modifications.
\let\qed\relax
\end{proof}

A natural question is to describe the above algebraic structures explicitly and perhaps to find a more succinct way of presenting them.

\section*{Acknowledgments}

The author would like to thank A.~Voronov for discussions on operads.

\section*{Declaration of interests}

The author does not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and has declared no affiliations other than their research organizations.

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