text:
\begin{document}
\title {\large{\bf Poisson-Nijenhuis Structure on Lie groupoids from the Invariance's Point of View}}
\vspace{2mm}
\author { \small{ \bf Gh. Haghighatdoost$^1$ }\hspace{-2mm}{\footnote{Corresponding author,  e-mail: gorbanali@azaruniv.ac.ir}} , \small{ \bf J. Ojbag$^2$}\hspace{-1mm}{ \footnote{ e-mail:j.ojbag@azaruniv.ac.ir}}  \\
{\small{$^{1,2}$\em Department
of Mathematics,Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran}}\\ }
 \maketitle

\begin{abstract}
In this paper, we introduce right-invariant (similarly, left-invariant) Poisson-Nijenhuis  Structures on Lie groupoids and their infinitesimal counterparts as called $(\Lambda , \mathbf{n})-$structures. Also, we present a mutual correspondence between $(\Lambda ,\mathbf{n})-$structures on Lie algebroids with Poisson-Nijenhuis structures $(\Pi , \mathbf{N})$ on their Lie groupoids under some conditions. Also, we will construct Poisson-Nijenhuis pair groupoids as an example from the invariance's point of view.
\end{abstract}

\section{Introduction}
Poisson-Lie groups introduced by Drinfel’d \cite{Drinfeld}. Recently many researchers working on geometric structures on Lie groupoids and try to extend known methods on Lie groups to Lie groupoids. By linearization a Lie groupoid at the units, one can correspond a Lie algebroid to it. Suppose that $G \rightrightarrows M$ be a Lie groupoid with source and target maps $s$ and $t$. We denote it's Lie algebroid by $AG$, equipped with anchor map $\rho$ and bracket $[.,.]$.\\
In section 2 we review some preliminary definitions and theorems about Poisson-Nijenhuis manifolds and groupoids. Moreover we remember some basic subjects about Lie groupoids and Lie algebroids (for more details refer to \cite{F0Magri},\cite{Kosmann},\cite{Kosman2},\cite{Kiril0Mackenzie}).\\
In section 3 we define right-invariant Poisson-Nijenhuis structure on Lie groupoid and infinitesimal counterpart of this, called algebraic structures corresponding to Poisson-Nijenhuis Lie groupoids.
Also, we prove that $(\Pi , \mathbf{N})$-structures on Lie groupoids are in one-to-one correspondence with $(\Lambda , n)$-structures on their Lie algebroids from the Invariance's Point of View.\\
In section 4 we construct right-invariant Poisson-Nijenhuis structure on the pair groupoids. The one-to-one correspondence between $(\Lambda , n)$-structures and right-invariant Poisson-Nijenhuis structure on the pair groupoids has been verified.
\section{Preliminaries}
\begin{definition}
A Lie groupoid consists of two smooth manifolds $G$ and $M$, together with a set of smooth morphisms $\{s, t, u, i, m\}$, namely:
\begin{itemize}
\item The source map $s: G \to M$, which is a surjective submersion;
\item The target map $t: G \to M$, which is a surjective submersion;
\item The unit map $u: M \to G$, which is an embedding and is denoted by $1_x := u(x)$ for $x \in M$. Each $u(x)$ is called a unit;
\item The inversion $i: G \to G$, which is a diffeomorphism and is denoted by $g^{-1}:= i(g)$ for $g \in G$;
\item The multiplication $m: G^2 \to G$ denoted by $(g, h) \to  g.h := m(g, h)$ is smooth, where $G^2=G _s\times_t G=\{(g,h) \in G\times G|s(g)=t(h)\}$.
\end{itemize}
These are subject to the conditions that:
\begin{itemize}

\item[1.] $s(gh)=s(h), t(gh)=t(g)$.
\item[2.] $(gh)k=g(hk)$, whenever $s(g)=t(h)$ and $s(h)=t(k)$.
\item[3.] $1_{t(g)}g=g=g1_{s(g)}$
\item[4.] $gg^{-1}=1_{t(g)}, \;g^{-1}g=1_{s(g)}$.

\end{itemize}
\end{definition}
The main difference between Lie groups and Lie groupoids is that for groups, we could compose any two elements, while for Lie groupoids
this is only possible if the first arrow ends where the second one starts.\\
If the base $M$ is a point, a Lie groupouid $G \rightrightarrows M$ is the Lie group.\\
Consider Lie groupoid $G \rightrightarrows M$, for all $x \in M$, $s^{-1}(x)$ is called its source-fibre or $s$-fibre, $G_x := s^{-1}(x) \cap t^{-1}(x)$ its isotropy group and $L_x := t(s^{-1}(x))$ its orbit. $L_x \subset M$ is an embedded submanifold of $G$.
$G$ is called transitive if it has only one orbit. Its orbit space is a single point. The pair groupoid $M \times M \rightrightarrows M$ is an important example of a transitive Lie groupoid.
\begin{definition}
For a Lie groupoid $G \rightrightarrows M$,
\begin{itemize}

 \item $G \rightrightarrows M$ source-connected ($s$-connected), if $s^{-1}(x)$ is connected for each $x \in M$,
 \item $G \rightrightarrows M$ source-simply-connected ($s$-simply connected), if $s^{-1}(x)$ is connected and simply-connected for each $x \in M$.

\end{itemize}
\end{definition}
\begin{definition}\label{def2}
Let Lie groupoids $G \rightrightarrows M ,\; G^\prime \rightrightarrows M^\prime$. A morphism between Lie groupoids is a pair of maps $F: G \to G^\prime \;,\;f:M \to M^\prime $ such that:
\begin{itemize}
\item
$s^\prime \circ F=f \circ s,\;\;t^\prime \circ F=f \circ t$,
\item
$F(hg)=F(h)F(g),\;\;\forall (h,g)\in G \ast G$.
\end{itemize}
If $F$ and (hence) $f$ are diffeomorphisms, the morphism of groupoids called isomorphism of Lie groupoids.
\end{definition}

\begin{definition}
A Lie algebroid is a vector bundle $A$ on base $M$ together with a bracket of sections $\Gamma A \times \Gamma A \to \Gamma A$ and a map $\rho: A \to TM$ such that

\begin{itemize}
\item the bracket of sections makes $\Gamma A$ an $R-$Lie algebra,
\item $[X,fY]=f[X,Y]+\rho (X)(f)Y, \forall X,Y \in \Gamma A, \; f\in C^\infty(M)$,
\item $\rho[X,Y]=[\rho X,\rho Y], \;\; X,Y\in \Gamma A$.
\end{itemize}
\end{definition}
For a Lie groupoid $G\rightrightarrows M$, restrict $TG$ to the identity elements; get $T_{1M}G$, a vector bundle on $M$. Right-translations $R_g$, map $s-$ fibers to $s-$ fibers. So take the kernel of $T(s): T_{1M}G \to TM$. Call this $AG$.
Each $X\in \Gamma AG$ defines a right-invariant vector field $\overrightarrow{X}$ on $G$ by $\overrightarrow{X}(g)=Xg$. That is, $\overrightarrow{X}$ is $s-$vertical and $\overrightarrow{X}(hg)=\overrightarrow{X}(h)g$ for all $h,g$.
Each right-invariant vector field is $\overrightarrow{X}$ for some $X\in \Gamma AG$. The bracket of right-invariant vector fields is right-invariant. Define bracket on $\Gamma AG$ by $\overrightarrow{[X , Y]}=[\overrightarrow{X},\overrightarrow{Y}]$.  $AG$ is the Lie algebroid of $G$.

\begin{definition}
Let $M$ be a smooth manifold and $N:TM \to TM$ be a vector valued $1-$form, or a $(1,1)-$tensor on $M$. Then its Nijenhuis torsion $\tau N$ is a vector valued $2-form$ deﬁned by
\[\tau N(X,Y):=[NX,NY ]-N([NX,Y]+[X,NY ]-N[X,Y ]), \; for X,Y\in \Gamma(TM).\]
An $(1,1)$-tensor $N$ is called a Nijenhuis tensor if its Nijenhuis torsion vanishes.
\end{definition}
Given a Nijenhuis tensor $N$, one can define a new Lie algebroid structure on $TM$ deformed by $N$. We denote this Lie algebroid by $(TM)_N$. The bracket $[ , ]_N$ and anchor $id_N$ of this deformed Lie algebroid are given by
\[[X,Y]_N=[NX,Y]+[X,NY ]-N[X,Y ]\; \text{and}\; id_N=N, \;\forall X,Y\in \Gamma(TM).\]
Let $M$ be a smooth manifold and $\Pi \in \Gamma (\wedge^2 TM)$ a bivector field on $M$. Then one can define a skew-symmetric bracket $[, ]_\Pi$ on the space $\Omega^1(M)$ of $1$-forms on $M$, given by
\[[\alpha,\beta]_\Pi :=\mathcal{L}_{\Pi^{\sharp}\alpha}\beta-\mathcal{L}_{\Pi^{\sharp}\beta}\alpha-d(\Pi(\alpha,\beta)), \forall \alpha,\beta \in \Gamma(T^{\ast}M),\] where $\Pi^{\sharp}:T^\ast M \to TM$, $\alpha \to \Pi(\alpha , -)$ is the bundle map induced by $\Pi$. If $\Pi$ is a Poisson bivector (that is, $[\Pi,\Pi]=0$), then the cotangent bundle $T^\ast M$ with the above bracket and the bundle map $\Pi^{\sharp}$ forms a Lie algebroid. We call this Lie algebroid as the cotangent Lie algebroid of the Poisson manifold $(M,\Pi)$ and is denoted by $(T^\ast M)_\Pi$. $(T^\ast M)_\Pi$ knows everything about $(M,\Pi)$ .

\begin{definition}\label{2.6}
A Poisson-Nijenhuis manifold is a manifold $M$ together with  a Poisson bivector $\Pi \in \Gamma(\wedge^2 TM)$ and a Nijenhuis tensor $N$ such that  they are compatible in the following senses:
\begin{itemize}
\item $N \circ \Pi^{\sharp}=\Pi^{\sharp} \circ N^\ast$ (thus, $N \circ \Pi^{\sharp}$ defines a bivector field $N \Pi$ on $M$),
\item $C(\Pi,N)\equiv 0$,
\end{itemize}
where
\[C(\Pi,N)(\alpha,\beta):=[\alpha,\beta]_{N\Pi}-([N^\ast \alpha,\beta]_\Pi+[\alpha,N^\ast \beta]_\Pi-N^\ast[\alpha,\beta]_\Pi),\; \text{for}\; \alpha, \beta \in \Omega^1(M).\]
The skew-symmetric $C^\infty (M)$-bilinear operation $C(\Pi,N)(-,-)$ on the space of $1$-forms is called the Magri-Morosi concomitant of the Poisson structure $\Pi$ and the Nijenhuis tensor $N$.\\
A Poisson-Nijenhuis manifold as above is denoted by the triple $(M,\Pi,N)$. If $\Pi$ is non-degenerate, that is, defines a symplectic structure $\omega$ on $M$, then $(M,\omega,N)$ is said to be a symplectic-Nijenhuis manifold.
\end{definition}

\begin{definition}
A Poisson groupoid is a Lie groupoid $G\rightrightarrows M$ equipped with a Poisson structure $\Pi$ on $G$, if $\Pi^{\sharp}:T^\ast G \to TG$ be a Lie groupoid morphism from the cotangent Lie groupoid $T^{\ast} G \rightrightarrows A^\ast G$ to the tangent Lie groupoid $TG \rightrightarrows TM$.

\end{definition}

\begin{definition}\label{defN}
Let $G \rightrightarrows M$ be a Lie groupoid. A multiplicative $(1,1)$-tensor $\mathbf{N}$ on the groupoid is a pair $(N,N_M)$ of $(1,1)$-tensors on $G$ and $M$, respectively, such that
\begin{center}
\begin{tikzcd}
  TG \arrow[to=2-1, shift right=0.65ex]\arrow[to=2-1, shift left=0.65ex]\arrow[to=1-2, "N"]
  & TG \arrow[to=2-2, shift right=0.65ex]\arrow[to=2-2, shift left=0.65ex] \\
   TM \arrow[to=2-2]\arrow[to=2-2, "N_M"]
  & TM

\end{tikzcd}
\end{center}
is a Lie groupoid morphism from the tangent Lie groupoid to itself. In this case, $N_M$ is the restriction of $N$ to the unit space $M$. Thus, $N_M$ is completely determined by $N$. Hence we may use $N$ to denote a multiplicative $(1,1)$-tensor.
\end{definition}

\begin{definition}
A Nijenhuis groupoid is a Lie groupoid $G\rightrightarrows M$ together with a multiplicative Nijenhuis tensor $\mathbf{N}$ (cf. Definition \ref{defN}). A Nijenhuis groupoid may also be denoted by $(G\rightrightarrows M,\mathbf{N})$.

\end{definition}

\begin{definition}
A Poisson-Nijenhuis groupoid is a Lie groupoid $G\rightrightarrows M$ together with a Poisson-Nijenhuis structure $(\Pi,N)$ on $G$ such that $(G\rightrightarrows M,\Pi)$ forms a Poisson groupoid and $(G\rightrightarrows M,N)$ a Nijenhuis groupoid. Thus, a Poisson-Nijenhuis groupoid is a Poisson groupoid $(G\rightrightarrows M,\Pi)$ together with a multiplicative Nijenhuis tensor $N:TG \to TG$ such that $(G,\Pi,N)$ is a Poisson-Nijenhuis manifold. A Poisson-Nijenhuis groupoid as above is denoted by the triple $(G\rightrightarrows M,\Pi,N)$.
\end{definition}

\section{Right-invariant Poisson-Nijenhuis structures on Lie groupoids}
In this section, we define Poisson-Nijenhuis Lie groupoids from the invariant point of view, and their infinitesimal counterpart on the Lie algebroids $AG$ of $G$.
\begin{definition}\label{defp}
A Poisson-Nijenhuis structure $(\Pi,\mathbf{N})$ on a Lie groupoid $G\rightrightarrows M$ is said to be right-invariant, if:
\begin{itemize}
\item[1.] The Poisson structure $\Pi$ is right invariant, i.e., there exists $\Lambda \in \Gamma(\wedge^2 AG)$ such that $\Pi=\overrightarrow{\Lambda}$.
\item[2.] Multiplicative $(1,1)-$tensor $\mathbf{N}=(N,N_M)$ also is right-invariant, i.e., there are linear endomorphisms $n:\Gamma(AG) \to \Gamma(AG)$ and $n_M:TM \to TM$ such that
\[N=\overrightarrow{n},\;\;N_M=\overrightarrow{n_M}\]
\end{itemize}

\end{definition}
In the following we prove our claims only for $N$, beacuse $N_M$ is completely determined by $N$. This is also true for $n$ and $n_M$.
\begin{proposition}\label{pro1}
Let $(\Pi,\mathbf{N})$ be a right-invariant Poisson-Nijenhuis structure on a Lie groupoid $G \rightrightarrows M$ with Lie algebroid $AG$ and space of unites $1_M \subset G$. If $\Lambda \in \Gamma(\wedge^2 AG)$ that are the values of $\Pi$ restricted to space of unites $1_M$ and $(N|_{AG},N_M|_{TM})=\mathbf{n}$, then
\begin{itemize}
\item[1.] $[\Lambda,\Lambda]_{SN}=0$, where $[,]_{SN}$ is the Schouten-Nijehuis bracket,
\item[2.] $\mathbf{n}$ is Nijenhuis operator on $AG$, i.e., $\tau n(X,Y)=\tau n_M(X,Y)=0,\;\forall X,Y \in \Gamma AG$,
\item[3.] $\mathbf{n} \circ \Lambda^\sharp=\Lambda^\sharp \circ \mathbf{n}^\ast$
\item[4.] The Magri-Morosi concomitant's $C(\Lambda,\mathbf{n})(\alpha,\beta)=0$,
\item[5.] $\overrightarrow{\Lambda}^\sharp$ and $\overrightarrow{\mathbf{n}}$ are Lie groupoid morphisms.
\end{itemize}

\end{proposition}
\begin{proof}

1. With respect to definition \ref{defp}, since $\Pi$ is a Poisson bivector and bracket on $\Gamma (AG)$ is defined by $\overrightarrow{[X,Y]}=[\overrightarrow{X},\overrightarrow{Y}]$, hence
\[[\Pi,\Pi]=0 \longrightarrow [\overrightarrow{\Lambda},\overrightarrow{\Lambda}]=\overrightarrow{[\Lambda,\Lambda}]=0\]
2. As we know, the Nijenhuis torsion is
\[\tau n(X,Y):=\overrightarrow{[N,N ](X,Y)}=[\overrightarrow{n},\overrightarrow{n}](X,Y)=[N,N](\overrightarrow{X},\overrightarrow{Y})=0, \; \;\;\forall X,Y\in \Gamma AG.\]
The proof is same for $n_M$.\\
3. Since $\Pi^\sharp(\overrightarrow{\alpha})=\overrightarrow{\Lambda^\sharp(\alpha)}$ and $N^\ast(\overrightarrow{\alpha})=\overrightarrow{n^\ast(\alpha)}$ for $\alpha \in T^\ast A$, we have\\
\[(N \circ \Pi^\sharp-\Pi^\sharp \circ N^\ast)(\overrightarrow{\alpha})=(\overrightarrow{n} \circ \overrightarrow{\Lambda^\sharp}-\overrightarrow{\Lambda^\sharp} \circ \overrightarrow{n^\ast})(\alpha)=\overrightarrow{(n \circ \Lambda^\sharp-\Lambda^\sharp \circ n^\ast)(\alpha)}\]
4. The Magri-Morosi concomitant's defined as $C(\Pi,\mathbf{N}):\Gamma(T^\ast G) \times \Gamma(T^\ast G) \to \Gamma(T^\ast G)$ . Consider the direct product Lie groupoid $T^\ast G \times T^\ast G \rightrightarrows A^\ast \times A^\ast$.\\
The Lie algebroid of Lie subgroupoid $T^\ast G \oplus_G T^\ast G \rightrightarrows A^\ast \oplus_M A^\ast$ is isomorphic to the Lie algebroid $T^\ast A \oplus_A T^\ast A \rightrightarrows A^\ast \oplus_M A^\ast$ \cite{Apurba0Das}. We can consider the Magri-Morosi concomitant as a Lie groupoid morphism from $T^\ast G \oplus_G T^\ast G \rightrightarrows A^\ast \oplus_M A^\ast$ to $T^\ast G \rightrightarrows A^\ast$, therefore we have
\[C(\Lambda ,\mathbf{n})=j^\prime_G \circ \mathcal{L}(C(\Pi,\mathbf{N})) \circ (j^\prime_G \oplus j^\prime_G)^{-1}:T^\ast A \oplus_A T^\ast A \to T^\ast A\]
Where $j: T(AG) \to A(TG)$ and $j^\prime: A(T^\ast G) \to T^\ast(AG)$ are canonical isomorphisms \cite{Kiril0Mackenzie2},\cite{Kiril0Mackenzie3}.\\
Since $C(\Pi ,\mathbf{N})=0$, therefore $C(\Lambda ,\mathbf{n})=0$.\\
According to the definitions \ref{defN} and \ref{defp} (5) is obviuos.
\end{proof}
\\
Similar to the case of lie algebras we can find a linear isomorphism between lie algebroid and tangent space of corresponding Lie groupoid.
\begin{lemma}\label{lem0}
Left invariant vector fields on Lie groupoid $G \rightrightarrows M$ identified with section of Lie algebroid $AG$.
\end{lemma}
\begin{proof}\cite{Kiril0Mackenzie}
\end{proof}
\\
Suppose $G \rightrightarrows M$ is a Lie groupoid with Lie algebroid $AG$ and $\Lambda \in \Gamma (\wedge^2 AG)$ is a bivector, $n:\Gamma(AG) \to \Gamma(AG)$ is a linear endomorphism on $AG$ that satisfies in (1-5) conditions of proposition \ref{pro1}. In the following theorem we proof that $(\overrightarrow{\Lambda},\overrightarrow{n})$ define a Poisson-Nijenhuis structure on Lie groupoid $G$. The $s-$connected and $s-$simply connected for our intended examples of Lie groupoids are necessary, beacuse of we need these to generate Lie groupoid $G \rightrightarrows M$ by flows of exponential map.
\begin{theorem}\label{theorem1}
Let $s-$connected and $s-$simply connected Lie groupoid $G \rightrightarrows M$ with Lie algebroid $AG$. Consider $ \Lambda \in \Gamma(\wedge^2AG)$ be an element  satisfying $ [\Lambda ,\Lambda]=0$. Then $\Pi =\overrightarrow{\Lambda}$ defines a Poisson groupoid structure on $G$.\\
Furthermor for the endomorphisms $n:\Gamma(AG) \to \Gamma(AG)$ and $n_M: TM \to TM$ there exists multiplicative $(1,1)$-tensors $N:TG \to TG,\;N_M:TM \to TM $ such that $ \overrightarrow{n}=N,\;\overrightarrow{n_M}=N_M $ and compatible with $\Pi$.
\end{theorem}
\begin{proof}
According to the definition of the bivector $\Pi$ we have
\[[\Pi,\Pi] = [\overrightarrow{\Lambda},\overrightarrow{\Lambda}]=\overrightarrow{[\Lambda ,\Lambda]}\]
Since $[\Lambda ,\Lambda]=0$, therefore $[\Pi,\Pi] =0$.\\
It only remains to show $\Pi^{\sharp}$ is a Lie groupoid morphism from the cotangent Lie groupoid $T^{\ast} G \rightrightarrows (AG)^{\ast}$ to the tangent Lie groupoid $TG \rightrightarrows TM$. From definition
\[\Pi^\sharp(\overrightarrow{\alpha})=\overrightarrow{\Lambda}^\sharp(\alpha)\]
and this fact that $\Lambda $ satisfy in conditions of proposition \ref{pro1}, hence according to part (5)  we have  $\overrightarrow{\Lambda}^\sharp$ is Lie groupoid morphism. The following diagram, show that $\Pi^\sharp$ is morphism of groupoids:
\begin{center}
\begin{tikzcd}
  T^\ast G \arrow[to=1-2, shift right=0.65ex]\arrow[to=1-2, shift left=0.65ex]\arrow[to=2-1, "\Pi^\sharp"]
  & A^\ast G \arrow[to=2-2, "\rho_\ast"] \\
   TG \arrow[to=2-2, shift right=0.65ex]\arrow[to=2-2, shift left=0.65ex]
  & TM

\end{tikzcd}
\end{center}
 Hence $(G, \Pi )$ is Poisson groupoid and this will complete the first part of the proof.\\
According to lemma \ref{lem0} every section of Lie algebroid $AG$ can be extended to a right-invariant vector field on Lie groupoid $G$. Hence the endomorphisms on the sections of Lie algebroid $AG$ can be extended to endomorphisms on tangent space $TG$ such that send right-invariant vector field to itself. Therefore we have $\overrightarrow{n}=N$ (Similarly, we have $ \overrightarrow{n_M}=N_M$).
We can define an isomorphism between sections of Lie algebroid and right invariant vector fields on $G$. We show this isomorphism by $\lambda_\alpha$. Therefore we have the commutative diagram as follow:

\begin{center}
\begin{tikzcd}
  \Gamma (AG) \arrow[to=2-1]\arrow[to=1-2, "n"]\arrow[to=2-1, "\lambda_\alpha"]
  & \Gamma (AG) \arrow[to=2-2]\arrow[to=2-2, "\lambda_\alpha"] \\
   TG \arrow[to=2-2, "N"]
  & TG

\end{tikzcd}
\end{center}
For expressing the compatibility of a pair $(\Pi , N)$, where $\Pi=\overrightarrow{\Lambda}$ is a Poisson bivector and $N=\overrightarrow{n}$ is multiplicative $(1,1)-$tensor, we verify  $N\Pi=\Pi N$ and $N\Pi^\sharp=\Pi^\sharp N^t$ or equivalently $N \Pi - \Pi N=0$ and $N \circ \Pi^\sharp-\Pi^\sharp \circ N^\ast=0$ as follow:
\[N \Pi - \Pi N=\overrightarrow{n} \overrightarrow{\Lambda} - \overrightarrow{\Lambda} \overrightarrow{n}=\overrightarrow{n\Lambda} -\overrightarrow{\Lambda n} \]
Since $n$ and $\Lambda$ are compatible, therefore $n\Lambda = \Lambda n$. Hence
\[\overrightarrow{n\Lambda} -\overrightarrow{\Lambda n}=\overrightarrow{\Lambda n}-\overrightarrow{\Lambda n}=0\]
Also we have
\[N \circ \Pi^\sharp-\Pi^\sharp \circ N^\ast=\overrightarrow{n} \circ \overrightarrow{\Lambda^\sharp}-\overrightarrow{\Lambda}\circ \overrightarrow{n}^\ast=\overrightarrow{n \circ \Lambda^\sharp}-\overrightarrow{\Lambda \circ n^\ast}\]
With respect to part 3 of proposition \ref{pro1} we have $n \circ \Lambda^\sharp=\Lambda \circ n^\ast$, therfore
\[\overrightarrow{n \circ \Lambda^\sharp}-\overrightarrow{\Lambda \circ n^\ast}=\overrightarrow{n \circ \Lambda^\sharp}-\overrightarrow{n \circ \Lambda^\sharp}=0\]
The concomitant Magri-Morosi $C(\Pi,N)(\overrightarrow{\alpha},\overrightarrow{\beta})=-\overrightarrow{C(\Lambda ,n)(\alpha , \beta)}$, hence $C(\Lambda ,n)=0$ gives $C(\Pi,N)=0$.
\end{proof}
$(\Lambda , n)$ which satisfy condition (1-5) of proposition \ref{pro1}; so-called \textit{$(\Lambda , n)-$ structure} on the Lie algebroid $AG$ and $(\overrightarrow{\Lambda},\overrightarrow{n})$ is right-invariant Poisson-Nijenhuis structure on $G$. similarly, we can define left-invariant Poisson-Nijenhuis structures. It is sufficient to replace right by left.

\section{Example}
In this section we consider the groupoid $G \rightrightarrows M$ with Lie algebroid $A$. As we know the tangent Lie groupoid of  the groupoid $G \rightrightarrows M$ is $TG \rightrightarrows TM$ whose structural maps are all obtained by taking the derivatives of the structural maps of $G$ and the cotangent bundle of a Lie groupoid $G \rightrightarrows M$ also carries a natural Lie groupoid structure, $T^\ast G \rightrightarrows A^\ast  $, where $A^\ast $ is the dual vector bundle to $A$. The source and target maps $\tilde{s}, \tilde{t}: T^\ast G \to A^\ast  $ are defined by the restriction of covectors to the subspaces tangent to the $s-$fibers and $t-$fibers, respectively \cite{Bursztyn}.\\
In this section, we define a right-invariant Poisson-Nijenhuis structure for Piar groupoids. Also, we show that there is a one-to-one correspondence between the Poisson-Nijenhuis Lie algebroid and corresponding Lie groupoid from the invariance's point of view.

\begin{example}
Any manifold $M$ gives rise to another Lie groupoid called the pair groupoid of $M$ which is indicated by  $M \times M \rightrightarrows M$ and a pair $(x, y)$ is seen as an arrow from $x$ to $y$ and the composition is given by:
\[ (x, y).(y, z)  = (x, z)\]
Clearly, one has $1_x = (x, x)$ and $(x, y)^{-1}=(y, x)$.\\
The Lie algebroid of the pair groupoid $M \times M \rightrightarrows M$ is $TM \to M$ with the usual Lie bracket of vector fields and the identity map as the anchor.\\
Let $\Pi$ be a Poisson bivector on Lie groupid $M\times  M \rightrightarrows M$. Since there are two functions
\[\Pi^\sharp:T^\ast(M\times M) \to T(M\times M),\;id:A^\ast  \to TM\] such that the diagram

\begin{center}
\begin{tikzcd}
  T^\ast(M\times M) \arrow[to=2-1, shift right=0.65ex]\arrow[to=2-1, shift left=0.65ex]\arrow[to=1-2, "\Pi^\sharp"]
  & T (M \times M) \arrow[to=2-2, shift right=0.65ex]\arrow[to=2-2, shift left=0.65ex] \\
   A^\ast \arrow[to=2-2]\arrow[to=2-2, "id"]
  & TM

\end{tikzcd}
\end{center}
is commutative. The existence of these functions concludes from this fact that says: for every Lie groupoid $ G \rightrightarrows M $, there is a natural morphism of Lie groupoids $ G \rightrightarrows M \times M, \; g \mapsto (s (g), t (g)) $. Also, every Poisson manifold $(M,\Pi)$ can be seen as a Lie algebroid $A$ with a vector bundle isomorphism $A \simeq T^\ast M$ satisfying certain properties \cite{Luca}.\\
Hence $\Pi^\sharp: T^\ast (M \times M) \to T(M \times M)$ is a Lie groupoid morphism between $T^\ast(M\times M) \rightrightarrows A^\ast $ and $T(M\times M) \rightrightarrows TM$, where $A^\ast $ is dual of Lie algebroid of $M\times M \rightrightarrows M$. Therefore the Lie groupoid $M \times M \rightrightarrows M$ is a Poisson groupoid.\\
The existence of Nijenhuis tensor on pair groupoid $M \times M \rightrightarrows M$ is obvious. Consider $N=id$ and $N_M$ consider the function that project the tangent space to the base space as follow:
\begin{center}
\begin{tikzcd}
  T^\ast(M\times M) \arrow[to=2-1, shift right=0.65ex]\arrow[to=2-1, shift left=0.65ex]\arrow[to=1-2, "id"]
  & T (M \times M) \arrow[to=2-2, shift right=0.65ex]\arrow[to=2-2, shift left=0.65ex] \\
   TM \arrow[to=2-2]\arrow[to=2-2, "Proj"]
  & M

\end{tikzcd}
\end{center}
Examining the terms of the definition \ref{def2} is simple.\\
Examining the conditiones of the definition \ref{2.6} are as follow:\\
Let $f \in T^\ast (M \times M)$, then we have:
\[(\mathbf{N} \circ \Pi^\sharp)(f)=\mathbf{N}(\Pi^\sharp(f))=(id(\Pi^\sharp(f)),Proj (\Pi^\sharp(f)))\]
On the other hand
\[(\Pi^\sharp \circ N^\ast)(f)=(\Pi^\sharp\circ \mathbf{N})(f)=\Pi^\sharp(\mathbf{N}(f))=\Pi^\sharp(id(f),Proj(f))\]
With respect to this fact that $\Pi^\sharp(f)$ and $\mathbf{N}$ are Lie group homomorphisms, we have:
\[\mathbf{N}\circ \Pi^\sharp = \Pi^\sharp \circ \mathbf{N}^\ast .\]
It is sufficient to show that $C(\Pi , \mathbf{N})=0$, this is true because of we consider $\mathbf{N}=(id,Proj)$ and usual bracket.\\
Hence $(M\times M \rightrightarrows M, \Pi, \mathbf{N})$ is Poisson-Nijenhuis Lie groupoid.\\
As we mentioned above the Lie algebroid of $M \times M \rightrightarrows M$ is $TM \to M$ with identity map as anchor and usual bracket, we use $A$ for it.\\
Now according to lemma \ref{lem0} we can consider $\Lambda \in \Gamma (A)$ and an arbitarary endomorphism such as $n$ on algebroid $A$ and extend these to the right (left) invariant vector field $\overrightarrow{\Lambda},\overrightarrow{n} $ on groupoid. With respect to theorem \ref{theorem1} $(\overrightarrow{\Lambda},\overrightarrow{n}) $ are Poisson-Nijenhuis structure on pair groupoid.
\end{example}

\bigskip

\providecommand{\bysame}{\leavevmode\hbox
to3em{\hrulefill}\thinspace}


text:
\section{Immersed maps and the cellular DGA}  \label{sec:4}

In \cite{RuSu1, RuSu2, RuSu25}, a cellular version of Legendrian contact homology is introduced and shown to be equivalent to the usual LCH DGA in the case of closed Legendrian surfaces.  In Sections \ref{sec:4-1} and \ref{sec:4-2}, we briefly review the cellular DGA and extend its definition to the case of (compact) Legendrian cobordisms.  The DGA of the identity  cobordisms is computed in Section \ref{sec:4-3} and shown to be a mapping cylinder DGA.   In Section \ref{sec:4-4} a cellular version, $F_\comp$, of the immersed LCH functor is defined with a domain category consisting of $\rho$-graded Legendrians equipped with some additional data and with compact Legendrian cobordisms as morphisms.
Finally, in Section \ref{sec:4-5}
we state in Proposition \ref{prop:Fisomorph} a precise relationship between the immersed and cellular LCH functors, $F$ and $F_{\comp}$, that will allow us to work with $F_{\comp}$ in place of $F$ when considering induced augmentation sets; see Corollary \ref{cor:augmentationset}.
Specifically, after unifying the domain categories by precomposing with suitable functors, $F$ and $F_{\comp}$ become isomorphic.
The proof of the isomorphism is an extension of the isomorphism between the cellular and LCH DGAs for closed Legendrian surfaces from \cite{RuSu1, RuSu2, RuSu25}, and the details are postponed until the Appendices \ref{app:iso} and \ref{app:A}.

\subsection{Review of the cellular DGA}  \label{sec:4-1} The cellular DGA construction requires as input a Legendrian knot or surface equipped with a suitable polygonal decomposition of its base projection.    Recall that for a generic $1$-dimensional Legendrian knot $\Lambda \subset J^1M$ the singularities of the front projection, $\pi_{xz}(\Lambda) \subset M \times \R$, are cusp points and crossings points (i.e. transverse double points).  Generically, front projections of  Legendrian surfaces have crossing arcs and cusp edges as codimension 1 singularities and triple points (intersection of three smooth sheets of $\Lambda$), cusp-sheet intersections (where a smooth sheet intersects a cusp edge), and swallowtail points as codimension 2 singularities.  See eg. \cite{ArnoldWave}; the front singularities for surfaces are illustrated in \cite[Section 2.2]{RuSu1}.  We say that $\Lambda$ has {\bf generic front and base projections} if the front singularities are generic, and the base projections (to $M$) of the different classes of front singularities are all self transverse and transverse to one another.

\begin{definition}  \label{def:decorated}
Let $\Lambda \subset J^1M$ be a closed Legendrian submanifold with $\dim \Lambda = 1$ or $2$ having generic front and base projections.
  A {\bf compatible polygonal decomposition} for $\Lambda$ is a polygonal\footnote{By polygonal decomposition, we mean a CW-complex decomposition such that the boundary of each $2$-cell consists of a sequence of vertices and edges (with repeats allowed).} decomposition, $\mathcal{E} = \{e^d_\alpha\}$, of the base projection of $\Lambda$,
\[
\sqcup_{d = 0}^2 \sqcup_\alpha e^d_\alpha  =  \pi_x(\Lambda)   \subset M,
\]
(where the superscript $0 \leq d \leq 2$ denotes the dimension of a cell) such that the base projection of the singular set of $\pi_{xz}(\Lambda)$ (crossings, cusps, swallow tail points, etc.) is contained in the $1$-skeleton of $\mathcal{E}$.
In addition, we require that:
\begin{enumerate}
\item Each $1$-cell is assigned an orientation.
\item Each $2$-cell is assigned an initial and terminal vertex, $v_0$ and $v_1$.  If $v_0 = v_1$, then a prefered direction around the boundary of the $2$-cell is also chosen.
\item At each swallowtail point, $s_0$, the two polygonal corners that border the crossing arc near its endpoint at $s_0$ are labelled as $S$ and $T$.
\end{enumerate}
\end{definition}

Let  $(\Lambda, \mathcal{E})$ be a pair consisting of a closed Legendrian knot or surface, $\Lambda \subset J^1M$, together with a choice of compatible polygonal decomposition.  The {\bf cellular DGA} of $(\Lambda, \mathcal{E})$ will be denoted $\alg(\Lambda, \mathcal{E})$ or $\calg(\Lambda, \mathcal{E})$, and is defined as follows.

\medskip

\noindent {\bf Algebra:}  Given a cell, $e^d_\alpha$, denote by $\{S^\alpha_p\}$ the set of {\bf sheets} of $\Lambda$ above $e^d_\alpha$.  By definition, sheets above $e^d_\alpha$ are those components of $\Lambda \cap \pi_x^{-1}(e^d_\alpha)$ not contained in any cusp edge.  (Note: (i) Sheets are subsets of $\Lambda$, not $\pi_{xz}(\Lambda)$, so that, eg., a crossing arc of $\pi_{xz}(\Lambda)$ above a $1$-cell corresponds to two sheets. (ii) A swallowtail point {\it is} considered to be a sheet above a $0$-cell.)  The algebra $\alg(\Lambda, \mathcal{E})$ is the free unital, associative $\Z/2$-algebra whose generators are in bijection with triples $(e^d_\alpha, S^\alpha_i, S^\alpha_j)$ where $S^\alpha_i$ and $S^\alpha_j$ are sheets above $e^d_\alpha$ such that the inequality $z(S^\alpha_i) > z(S^\alpha_j)$ holds pointwise above $e^d_\alpha$.  We denote the generator associated to $(e^d_\alpha, S^\alpha_i, S^\alpha_j)$  as
\[
a^\alpha_{i,j}, \quad b^\alpha_{i,j}, \quad \mbox{or} \quad  c^\alpha_{i,j}
\]
 when the dimension of $e^d_\alpha$ is $0$, $1$, or $2$ respectively.

\medskip

\noindent{\bf Grading:}
 A $\Z/\rho$-grading on $\mathcal{A}(\Lambda, \mathcal{E})$ arises from a choice of $\Z/\rho$-valued Maslov potential, $\mu$, for $\Lambda$.
  The $\Z/\rho$-grading of generators is
 \[
|a^\alpha_{i,j}| = \mu(S^\alpha_i) - \mu(S^\alpha_j)-1, \quad |b^\alpha_{i,j}| = \mu(S^\alpha_i) - \mu(S^\alpha_j),  \quad |c^\alpha_{i,j}| = \mu(S^\alpha_i) - \mu(S^\alpha_j)+1.
\]
(If $S^\alpha_i$ is a swallowtail point above a $0$-cell, then take $\mu(S^\alpha_i)$ to be the value of $\mu$ on the two sheets that cross near $S^\alpha_i$.)

\medskip

\noindent{\bf Differential:}  The differential $\partial: \mathcal{A}(\Lambda, \mathcal{E}) \rightarrow \mathcal{A}(\Lambda, \mathcal{E})$ is characterized on the generators of $\mathcal{A}(\Lambda, \mathcal{E})$ by matrix formulas whose precise form depends on the dimension of the associated cell $e^d_\alpha \in \mathcal{E}$.  We review these formulas here \emph{in the case that $\Lambda$ does not have swallowtail points}.  See also \cite{RuSu1, RuSu3}.

\begin{itemize} \item {\it For $0$-cells:}  We choose a linear ordering of the sheets of $\Lambda$ above $e^0_\alpha$ so that the $z$-coordinates appear in non-increasing order
\[
z(S^\alpha_{\iota(1)}) \geq z(S^\alpha_{\iota(2)}) \geq \cdots \geq z(S^\alpha_{\iota(n)})
\]
and use it to place the generators, $a^\alpha_{p,q}$, into a matrix, $A$, whose $(i,j)$-entry is $a^\alpha_{\iota(i), \iota(j)}$ when $z(S^\alpha_{\iota(i)}) > z(S^\alpha_{\iota(j)})$ and is $0$ otherwise.  When $\partial$ is applied entry-by-entry to $A$ we have
\[
\partial A = A^2.
\]

\item {\it For $1$-cells:}  After a choice of linear ordering of sheets as above, we place the generators, $b^\alpha_{p,q}$, associated to a $1$-cell, $e^1_\alpha$, into an $n \times n$ matrix, $B$.  In addition, we form $n\times n$ {\bf boundary matrices}, $A_-$ and $A_+$, associated to the initial and terminal vertices, $e^0_-$ and $e^0_+$,  of $e^1_\alpha$.  The sheets above $e^0_\pm$ are identified with a subset of the sheets above $e^1_\alpha$ (since every sheet of $e^0_\pm$ belongs to the closure of a unique sheet of $e^1_\alpha$).  Using this identification, we place the $e^0_\pm$ generators, $a^\pm_{p,q}$, into the corresponding rows and columns of the $n\times n$ matrices $A_\pm$.  Whenever two sheets of $e^1_\alpha$, $S^\alpha_k$ and $S^\alpha_{k'}$, meet at a cusp point above $e^0_\pm$, we place a $2 \times 2$ block of the form
\[
N = \left[ \begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array} \right]
\]
on the diagonal at the location of the two (possibly non-consecutive) rows and columns of $A_\pm$ that correspond to $S^\alpha_k$ and $S^\alpha_{k'}$ with respect to the linear ordering of sheets of $e^1_\alpha$.  All other entries of $A_\pm$ are $0$.  The differential on the $b^\alpha_{p,q}$ satisfies
\[
\partial B = A_+ (I+B) +(I+B) A_-.
\]

\item {\it For $2$-cells:}  The sheets above a $2$-cell, $e^2_\alpha$, are already linearly ordered by descending $z$-coordinate, and using this ordering we place the generators $c^\alpha_{i,j}$ into an $n\times n$-matrix, $C$.  We form $n\times n$ boundary matrices, $A_{v_0}$ and $A_{v_1}$, associated to the initial and terminal vertices, $v_0$ and $v_1$, for $e^2_\alpha$ following the same procedure as in the $1$-cell case.  Additional boundary matrices $B_1, \ldots, B_l$ are associated to the $1$-cells that appear around the boundary of $C$.  We assume that the numbering is such that $B_1, \ldots, B_{j}$ (resp. $B_{j+1}, \ldots, B_l$) are the boundary matrices for the sequence of edges that appear along the path $\gamma_{+}$ (resp. $\gamma_-$) where $\gamma_\pm$ are the two paths that travel around the boundary of $e^2_\alpha$ (in the domain of the characteristic map) from $v_0$ to $v_1$.  When $v_0=v_1$ one of $\gamma_\pm$ is constant (as specified by the choice of preferred direction around $\partial e^2_\alpha$).  The $B_i$ are formed using the same procedure as for the boundary matrices associated to vertices except that the $2\times 2$ $N$ blocks that correspond to pairs of cusp sheets are replaced with $2\times 2$ blocks of $0$.  The differential satisfies
\[
\partial C = A_{v_1} C + C A_{v_0} + (I+B_{j})^{\eta_j} \cdots (I+B_1)^{\eta_1} +(I+B_l)^{\eta_l} \cdots (I+B_{j+1})^{\eta_{j+1}}
\]
where $\eta_i$ is $+1$ (resp. $-1$) when the orientation of the $B_i$ edge agrees (resp. disagrees) with the orientation (from $v_0$ to $v_1$) of the corresponding path $\gamma_{\pm}$.  Note that since the $B_i$ are strictly upper triangular, $(I+B_{i})^{-1} = I +B_i + B_i^2 + \cdots +B_i^{n-1}$.

\end{itemize}

\begin{remark}
To define $\partial$ in the case that $\Lambda$ has swallowtail points, the following additions should be made.
\begin{itemize}
\item The definition of the boundary matrices $A_{\pm}$, $A_{v_0}$, and $A_{v_1}$ needs to be adjusted when the vertex is a swallowtail point of $\Lambda$.
\item Whenever a $2$-cell contains one of the corners labeled $S$ or $T$ at a swallowtail point, an additional matrix need to be inserted into the product $(I+B_j) \cdots (I+B_1)$ or $(I+B_{l}) \cdots (I+B_{j+1})$.
\end{itemize}
These details may be found in \cite{RuSu1} or \cite{RuSu3} and are not needed for the arguments that follow.
\end{remark}

\begin{observation} \label{obs:cobord}
Whenever $\mathcal{E}' \subset \mathcal{E}$ is a (CW) sub-complex, the collection of generators associated to cells of $\mathcal{E}'$ form a sub DGA of $\calg(\Sigma, \mathcal{E})$.  Moreover, if $\mathcal{E}'$ is a decomposition of a curve or curve segment $C \subset M$ such that the restriction of
$\Sigma$ to $C$ (as in Section \ref{sec:2-3}) is a $1$-dimensional Legendrian knot $\Lambda \subset J^1C$, then this sub-DGA is precisely $\calg(\Lambda, \mathcal{E}')$.
\end{observation}

\subsection{The cellular DGA for compact Legendrian cobordisms}  \label{sec:4-2}
In extending the definition of the cellular DGA to Legendrian cobordisms, it is most natural to consider compact cobordisms rather than conical cobordisms.  The following definition coincides precisely with a standard notion of Legendrian cobordism  introduced by Arnold, cf. \cite{Arnold1, Arnold2, ArnoldWave}.

\begin{definition} Given an interval $I \subset [0,1]$ and $\Lambda \subset J^1M$, we write $j^1(1_I \cdot \Lambda)$ for the {\bf Legendrian cylinder} on $\Lambda$ in $J^1(I \times M)$.  It is defined to be the product of $\Lambda \subset J^1M$ with the $0$-section in $T^*I$, i.e.
\begin{equation}  \label{eq:1I}
j^1(1_I \cdot \Lambda) = 0_{T^*I} \times \Lambda \subset T^*I \times J^1M \cong J^1(I \times M).
\end{equation}
The notation is chosen to be consistent with that of Definition \ref{def:j1} with $1_I:I \rightarrow \R_{>0}$ denoting the constant function $1$.

Let $\Lambda_0, \Lambda_1 \subset J^1M$.  A {\bf compact Legendrian cobordism} from $\Lambda_0$ to $\Lambda_1$, written $\Sigma: \Lambda_0 \rightarrow \Lambda_1$, is a Legendrian surface $\Sigma \subset J^1([0,1]\times M)$ that, for some $\epsilon >0$, $\Sigma$ agrees with the Legendrian cylinder
$j^1(1_{[0, \epsilon]} \cdot \Lambda_0)$ in $J^1([0, \epsilon] \times M)$ and agrees with $j^1(1_{[1-\epsilon, 1]} \cdot \Lambda_1)$ in $J^1([1-\epsilon, 1], \times M)$.
\end{definition}

When $\Sigma: \Lambda_0 \rightarrow \Lambda_1$ is a compact Legendrian cobordism, we modify the definition of compatible polygonal decomposition for $\Sigma$ to require that $\pi_x(\Sigma) \cap \left(\{0\} \times M\right)$ and $\pi_x(\Sigma) \cap \left( \{1\} \times M \right)$ are (CW) sub-complexes of $\mathcal{E}$, that we denote $\mathcal{E}_0$ and $\mathcal{E}_1$.  Then, the definition of the cellular DGA extends immediately to give DGAs $\mathcal{A}(\Sigma, \mathcal{E})$ when $\Sigma$ is a compact Lagrangian cobordism.  Moreover, since $\mathcal{E}_0$ and $\mathcal{E}_1$ may be viewed as polygonal decompositions for $\Lambda_0$ and $\Lambda_1$, as in the Observation \ref{obs:cobord}, we have inclusion maps
\begin{equation} \label{eq:i0i1}
i_0:\mathcal{A}(\Lambda_0, \mathcal{E}_0) \rightarrow \mathcal{A}(\Sigma, \mathcal{E}) \quad \mbox{and} \quad i_1:\mathcal{A}(\Lambda_1, \mathcal{E}_1) \rightarrow \mathcal{A}(\Sigma, \mathcal{E}).
\end{equation}

\subsection{The DGA of a product cobordism} \label{sec:4-3} Given a $1$-dimensional Legendrian with compatible polygonal decomposition, $(\Lambda, \mathcal{E})$, we now compute the DGA of the product cobordism $\Sigma = j^1(1_{[0,1]} \cdot \Lambda) \subset J^1([0,1] \times M)$. This will be a useful ingredient in a few later arguments.

\begin{figure}

\quad

\quad

\quad

\labellist
\small
\pinlabel $e^0_\alpha$ [l] at 12 166
\pinlabel $e^1_\beta$ [l] at 12 84
\pinlabel $\{0\}{\times}e^0_\alpha$ [r] at 190 164
\pinlabel $\{1\}{\times}e^0_\alpha$ [l] at 308 164
\pinlabel $(0,1){\times}e^0_\alpha$ [b] at 238 172
\pinlabel $\{0\}{\times}e^1_\beta$ [r] at 186 62
\pinlabel $\{1\}{\times}e^1_\beta$ [l] at 308 62
\pinlabel $(0,1){\times}e^1_\beta$ at 250 62
\pinlabel $v_0$ [r] at 190 8
\pinlabel $v_1$ [l] at 304 112

\endlabellist

\centerline{\includegraphics[scale=.6]{Images/Eprime2}}

\caption{Notation for cells in $\mathcal{E}$ (left) and $\mathcal{E}'$ (right).
}
\label{fig:Eprime2}
\end{figure}

As a preliminary, use $\mathcal{E}$ to form the {\bf product decomposition}, $\mathcal{E}'$, for $\pi_x(\Sigma) = [0,1] \times \pi_x(\Lambda)$ as follows:
 For each $d$-cell $e^d_\alpha \in \mathcal{E}$ (here, $d=0$ or $1$), decompose
\[
[0,1] \times e^d_\alpha = \{0\} \times  e^d_\alpha \sqcup (0,1) \times e^d_\alpha \sqcup \{1\} \times e^d_\alpha
\]
 so that  $\{k\} \times e^d_\alpha$ are $d$-cells in $\mathcal{E}'$, for $k=0,1$, and $(0,1) \times e^d_\alpha $ is a $(d+1)$-cell in $\mathcal{E}'$.   Orient $1$-cells of the form $\{k\} \times e^1_\beta$ in the same direction as $e^1_\beta$, and those of the form $(0,1) \times e^0_\alpha$ using the standard orientation of $(0,1)$.  For each $2$-cell, $(0,1) \times e^1_\beta$ choose the initial and terminal vertices $v_0$ and $v_1$ to be $\{0\} \times e^0_-$ and $\{1\} \times e^0_+ $ where $e^0_-$ and $e^0_+$ are the initial and terminal vertices of $e^1_\beta$.  See Figure \ref{fig:Eprime2}.

 Let us fix notation for the generators of the cellular DGA $\mathcal{A}(\Sigma, \mathcal{E}')$.  Observe that for any $e^d_\alpha \in \mathcal{E}$, the sheets of $\Lambda$ above $e^d_\alpha$ are in bijection with the sheets of $\Sigma$ above any of the $\{k\} \times e^d_\alpha$ or $(0,1) \times e^d_\alpha$ cells of $\mathcal{E}'$, so the generators associated to these cells are also in bijection with those of $e^d_\alpha$.
 \begin{itemize}
 \item For a $0$-cell $e^0_\alpha \in \mathcal{E}$ with corresponding generators $a^\alpha_{i,j}$ we notate the generators corresponding to  $\{0\} \times e^0_\alpha $, $\{1\} \times e^0_\alpha$, and $(0,1) \times e^0_\alpha$ as $i_0(a^\alpha_{i,j})$, $i_1(a^\alpha_{i,j})$, and $b^\alpha_{i,j}$.
\item For a $1$-cell $e^1_\beta \in \mathcal{E}$ with corresponding generators $b^\beta_{i,j}$ we notate the generators corresponding to  $\{0\} \times e^1_\beta$, $\{1\} \times e^1_\beta$, and $(0,1) \times e^1_\beta$ as $i_0(b^\beta_{i,j})$, $i_1(b^\beta_{i,j})$, and $c^\beta_{i,j}$.
\end{itemize}

In Proposition \ref{prop:alg01}, the DGA of the product cobordism $\Sigma$ is described as a mapping cylinder DGA.  We now briefly review the relevant definitions, referring the reader to \cite[Section 2]{PanRu1} for a more thorough treatment in the present algebraic setting of triangular DGAs over $\Z/2$.  Let $f:(\mathcal{A}, \partial_\mathcal{A}) \rightarrow (\mathcal{B}, \partial_\mathcal{B})$ be a DGA map between based DGAs.  The {\bf standard mapping cylinder DGA} of $f$ is $(\mathcal{A} * \widehat{\mathcal{A}} * \mathcal{B}, \partial_\Gamma)$ where $\widehat{\mathcal{A}}$ has generators $\{\widehat{a_i}\}$ in correspondence with the generators $\{a_i\}$ of $\mathcal{A}$ but with the degree shift $|\widehat{a_{i}}| = |a_i|+1$.  The
differential, $\partial_{\Gamma}$, satisfies
\[
\partial_{\Gamma}|_{\mathcal{A}} = \partial_{\mathcal{A}}, \quad \partial_{\Gamma}|_{\mathcal{B}} = \partial_{\mathcal{B}}, \quad \mbox{and} \quad \partial_\Gamma(\widehat{a_i}) = f(a_i) + a_i+ \Gamma \circ \partial_\mathcal{A}(a_i)
\]
where $\Gamma: \mathcal{A} \rightarrow \mathcal{A} * \widehat{\mathcal{A}} * \mathcal{B}$ is the unique $(f,\mathit{id}_\mathcal{A})$-derivation satisfying $\Gamma(a_i) = \widehat{a_i}$.

\begin{proposition}  \label{prop:alg01}  Given $(\Lambda, \mathcal{E})$ with $\Lambda \subset J^1M$ and $\dim \Lambda = 1$, let $(\Sigma, \mathcal{E}')$ be the product cobordism, $\Sigma = j^1(1_{[0,1]}\cdot \Lambda) \subset J^1([0,1] \times M)$ equipped with the product decomposition, $\mathcal{E}'$.
\begin{enumerate}
\item For $k=0,1$, the maps $i_k:\alg(\Lambda, \mathcal{E}) \rightarrow \alg(\Sigma, \mathcal{E}')$ that extend the correspondence of generators are DGA isomorphisms from $\alg(\Lambda, \mathcal{E})$ onto the sub-DGAs $\mathcal{A}_k \subset \alg(\Sigma, \mathcal{E}')$ associated to the subcomplexes $ \{k\} \times \mathcal{E}\subset \mathcal{E}'$.
\item Identifying the sub-algebra of $\alg(\Sigma, \mathcal{E}')$ generated by the $b^\alpha_{i,j}$ and $c^\beta_{i,j}$ with $\widehat{\mathcal{A}_0}$ using the grading preserving bijection $\widehat{i_0(a^\alpha_{i,j})} \leftrightarrow b^\alpha_{i,j}$ and $\widehat{i_0(b^\beta_{i,j})} \leftrightarrow c^\beta_{i,j}$
gives a DGA isomorphism
\[
\varphi:(\alg(\Sigma, \mathcal{E}'), \partial) \stackrel{\cong}{\rightarrow} (\alg_0 * \widehat{\alg_0} * \alg_1, \partial_\Gamma)
\]
with the standard mapping cylinder DGA of the map $i_1 \circ i_0^{-1}: \mathcal{A}_0 \rightarrow \mathcal{A}_1$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) is obvious. For (2), $\varphi$ is clearly an algebra isomorphism which is the identity on the $\mathcal{A}_k$, so we just to check that $\partial_\Gamma  \circ \varphi (x) = \varphi \circ \partial (x)$ when $x = b^\alpha_{i,j}$ or $c^\beta_{i,j}$.  From the definition of $\partial_\Gamma$,
 we have
\begin{align}
\label{eq:align1} \partial_\Gamma \widehat{i_0(a^\alpha_{i,j})}  & = i_1(a^\alpha_{i,j}) +i_0(a^{\alpha}_{i,j}) + \Gamma \circ \partial_\Lambda a^\alpha_{i,j}  \\
\label{eq:align2} \partial_\Gamma \widehat{i_0(b^\beta_{i,j})}  & = i_1(b^\beta_{i,j}) +i_0(b^\beta_{i,j}) + \Gamma \circ \partial_\Lambda b^\beta_{i,j}
\end{align}
where $\partial_\Lambda$ is the differential on $\alg(\Lambda, \mathcal{E})$ and $\Gamma:\alg(\Lambda, \mathcal{E}) \rightarrow \alg_1 * \widehat{\alg_0} * \alg_0$ is the $(i_1, i_0)$-derivation satisfying $\Gamma(x) = \widehat{i_0(x)}$ on generators.

Using matrix notation, this allows us to compute
\begin{align*}
\varphi \circ \partial( B_\alpha) &= \varphi\left(i_1(A_\alpha)(I+B_\alpha) + (I+B_\alpha) i_0(A_\alpha)\right) \\
 & =  i_1(A_\alpha)  + i_1(A_\alpha) \widehat{i_0(A_\alpha)} + i_0(A_\alpha) + \widehat{i_0(A_\alpha)} i_0(A_\alpha) \\
 & = i_1(A_\alpha) + i_0(A_\alpha) +  \Gamma( A_\alpha^2) \\
 & = \partial_\Gamma( \widehat{i_0(A_\alpha)}) = \partial_\Gamma \circ \varphi(B_\alpha),
\end{align*}
and
\begin{align*}
\partial_\Gamma \circ \varphi(C_\beta) & = \partial_\Gamma( \widehat{i_0(B_\beta)}) = i_1(B_\beta) + i_0(B_\beta) + \Gamma(\partial B_\beta) \\
 & = i_1(B_\beta) + i_0(B_\beta) + \Gamma(A_+(I+ B_\beta) + (I+B_\beta) A_-) \\
 & = i_1(B_\beta) + i_0(B_\beta) + \Gamma(A_+)(I+ i_0(B_\beta)) +i_1(A_+) \widehat{i_0(B_\beta)} \\
 &  \quad + \widehat{i_0(B_\beta)} i_0(A_-) + (I+ i_1(B_\beta)) \Gamma(A_-) \\
 & = \varphi\bigg(  i_1(B_\beta) + i_0(B_\beta) + B_+(I+ i_0(B_\beta))  +i_1(A_+) C_\beta  \\
 & \quad + C_\beta i_0(A_-) + (I+i_1(B_\beta)) B_-\bigg) \\
 & = \varphi\bigg( i_1(A_+) C_\beta+  C_\beta i_0(A_-) + (I+B_+)(I+ i_0(B_\beta))  + (I+i_1(B_\beta))(I+ B_-)\bigg)  \\
 & = \varphi \circ \partial(C_\beta).
\end{align*}
At the fourth equality, it should be observed that when $A_-$ and $A_+$ are the boundary matrices for $e^1_\beta$ associated to the vertices $e^0_-$ and $e^0_+$, the boundary matrices $B_-$ and $B_+$ for $(0,1) \times e^1_\beta$ associated to the edges $(0,1) \times e^0_-$ and $(0,1) \times e^0_+$ indeed satisfy $\varphi(B_\pm) = \Gamma(A_\pm)$.  [Note that when a $\left[\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}\right]$ block appears on the diagonal of $A_\pm$ due to two sheets of $e^1_\beta$ meeting at a cusp above $e^0_\pm$, since $\Gamma(1) =0$ (any derivation has this property), the appropriate $\left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right]$ block will appear in $\Gamma(A_\pm)$.]
\end{proof}

\subsection{Immersed DGA maps from cobordisms via the cellular DGA} \label{sec:4-4}  Our aim is to now define a cellular version of the immersed LCH functor  $F: \mathfrak{Leg}^\rho_{im} \rightarrow \mathfrak{DGA}^\rho_{\mathit{im}}$ from Corollary \ref{cor:LCHfunctor}.  Recall that the domain category for $F$ has $\rho$-graded Legendrians in $J^1M$ equipped with
regular metrics as objects and has (conical Legendrian isotopy classes of) conical Legendrian cobordisms in $J^1(\R_{>0}\times M)$ (equivalently, good immersed Lagrangian cobordisms in $\mathit{Symp}(J^1M)$) as morphisms.  For the cellular construction we instead work with a  category of compact cobordisms.

With $M = \R$ or $S^1$ and $\rho\geq 0$ fixed, define a {\bf cellular Legendrian cobordism category}, denoted $\mathfrak{Leg}^\rho_{\mathit{cell}}$, whose objects are triples $(\Lambda, \mathcal{E}, \mu)$ consisting of a $1$-dimensional Legendrian link, $\Lambda \subset J^1M$, together with a choice, $\mathcal{E}$, of compatible polygonal decomposition, and a choice, $\mu$, of $\Z/\rho$-valued Maslov potential.  Morphisms from $(\Lambda_0, \mathcal{E}_0, \mu_0)$ to $(\Lambda_1, \mathcal{E}_1, \mu_1)$
are equivalence classes of
compact Legendrian cobordisms $\Sigma \subset J^1([0,1] \times M)$ from $\Lambda_-$ to $\Lambda_+$ having generic base and front projection and equipped with a $\Z/\rho$-valued Maslov potential extending $\mu_0$ and $\mu_1$.  Here, two cobordisms are considered equivalent if their front and base projections are  {\bf combinatorially equivalent}.  That is, $\Sigma$ and $\Sigma'$ are equivalent if there is a homeomorphism $\phi:([0,1]\times M) \times \R \rightarrow ([0,1] \times M) \times \R$ with $\phi(\pi_{xz}(\Sigma)) = \phi(\pi_{xz}(\Sigma'))$ that is a composition  of (i) a  homeomorphism, $\phi_1$, of the $[0,1] \times M$ factor, and (ii) a homeomorphism, $\phi_2$, that preserves the $[0,1] \times M$ factor.  Moreover, $\phi_1$ and $\phi_2$ should be isotopic to the identity and equal  to the identity in a neighborhood of the boundary.

\begin{remark}
\begin{enumerate}
\item  As with the category $\mathfrak{Leg}^\rho_{\mathit{im}}$, in the definition of morphisms $\Sigma$ is not equipped with any additional structure beyond a choice of Maslov potential, eg. $\Sigma$ is not equipped with a polygonal decomposition.

 \item In contrast to $\mathfrak{Leg}^\rho_{\mathit{im}}$, we do NOT  allow general Legendrian isotopies of $\Sigma$ in the equivalence relation used to define morphisms.  This is because from the initial definition of our cellular LCH functor we will not check directly that the assignment of immersed DGA maps to Legendrian cobordisms  factors through general Legendrian isotopies.   However, it will later be established in Corollary \ref{cor:LegInv}  that the induced immersed DGA maps (considered up to immersed homotopy) are indeed Legendrian invariants of $\Sigma$.
 \end{enumerate}
 \end{remark}
 The next proposition defines the {\bf cellular LCH functor}, $F_{\mathit{cell}}$.
\begin{proposition} \label{prop:4-7}
There is a well-defined contravariant functor $F_{\mathit{cell}}: \mathfrak{Leg}^\rho_{\mathit{cell}} \rightarrow \mathfrak{DGA}^\rho_{im}$ given by
\[
\begin{array}{lcl}
(\Lambda, \mathcal{E}) & \leadsto & \calg(\Lambda, \mathcal{E}), \\
\Sigma: (\Lambda_0, \mathcal{E}_0) \rightarrow (\Lambda_1, \mathcal{E}_1) & \leadsto & \big[\calg(\Lambda_1, \mathcal{E}_1) \stackrel{i_1}{\rightarrow} \calg(\Sigma, \mathcal{E}') \stackrel{i_0}{\hookleftarrow} \calg(\Lambda_0, \mathcal{E}_0)\big] \end{array}
\]
where $\mathcal{E}'$ is any choice of  compatible polygonal decomposition for $\Sigma$ that restricts to $\mathcal{E}_{k}$ on $\{k\}\times M$, $k=0,1$, and the maps $i_0$ and $i_1$ are as in (\ref{eq:i0i1}).
\end{proposition}

\begin{proof}
To see that $F_{\mathit{cell}}$ is well-defined it is enough to verify independence of the choice of $\mathcal{E}'$.  (Since modifying $(\Sigma, \mathcal{E}')$ in a manner that preserves the combinatorics of the front and base projections does not affect the cellular DGA.)

In \cite[Section 4.2-4.4]{RuSu1}, it is shown in the case when $\Sigma$ is a closed surface that any  compatible polygonal decompositions $\mathcal{E}'$ and $\mathcal{E}''$ for $\Sigma$ can be made the same after some sequence of the following modifications and their inverses (see \cite{RuSu1} for the precise meanings).
\begin{enumerate}
\item Subdivide a $1$-cell.
\item Subdivide a $2$-cell.
\item Delete a $1$-valent edge.
\item Switch the $S$ and $T$ decorations at a swallowtail point.
\end{enumerate}
The proof extends with minor adjustments to the case of a compact cobordism.  Moreover, if $\mathcal{E}'$ and $\mathcal{E}''$ agree above $\partial([0,1]\times M)$, then the polygonal decomposition of $\partial([0,1] \times M)$ can be left unchanged throughout the sequence of modifications.  [To see this, follow the proof of \cite[Theorem 4.1]{RuSu1}, but treat all $0$-cells  (resp. $1$-cells) in $\partial([0,1] \times M)$ as if they belong to what is notated there as $\Sigma_2$ (resp. $\Sigma_1$).]

If $\mathcal{E}'$ and $\mathcal{E}''$ are related by one of the modifications (1)-(4), then \cite[Theorems 4.2-4.5]{RuSu1} provide  stable tame isomorphisms $\varphi:\calg(\Sigma, \mathcal{E}')*S' \rightarrow \calg(\Sigma, \mathcal{E}'')*S''$.  Moreover, in all cases $\varphi$ can be seen to restrict to the identity on the sub-algebras $\calg(\Lambda_{k}, \mathcal{E}_{k})$, $k=0,1$.  [For (1)-(3), $\varphi$ is defined using the construction of Proposition \ref{prop:quotient}, and generators from $\calg(\Lambda_{k}, \mathcal{E}_{k})$ are never among the generators that are canceled.  For (4), $\varphi$ is the identity on all generators except for $1$-cells with endpoints at swallowtail points.]   Thus, we will have identities $\varphi \circ i_{k} = i_{k}$, $k=0,1$, so that $\varphi$ provides the required immersed DGA homotopy.

To verify that $F_{\mathit{cell}}$ preserves composition, suppose that $\mathcal{E}'$ and $\mathcal{E}''$ are compatible polygonal decompositions for a pair of composable (compact) cobordisms, $\Sigma': (\Lambda_0, \mathcal{E}_0) \rightarrow (\Lambda_1, \mathcal{E}_1)$ and $\Sigma'':(\Lambda_0, \mathcal{E}_0) \rightarrow (\Lambda_1, \mathcal{E}_1)$.  Since $\mathcal{E}'$ and $\mathcal{E}''$ agree with $\mathcal{E}_1$ along $\Lambda_1$, they can be glued to form a well-defined compatible polygonal decomposition, $\mathcal{E}' \cup \mathcal{E}''$, on $\Sigma'' \circ \Sigma'$.  Moreover, when this decomposition is used for computing it is readily verified from Definition \ref{def:immersedcomp} that $F_{\mathit{cell}}( \Sigma'' \circ \Sigma') = F_{\mathit{cell}}( \Sigma') \circ  F_{\mathit{cell}}( \Sigma'')$.

Finally, we check that $F_{\mathit{cell}}$ preserves identities.  The identity morphism in $\mathit{Hom}_{\mathfrak{Leg}^\rho_{\mathit{cell}}}((\Lambda, \mathcal{E}), (\Lambda, \mathcal{E}))$ is the product cobordism, $\Sigma = j^1(1_{[0,1]} \cdot \Lambda)$, and for computing $F_{\mathit{cell}}(\Sigma)$ we equip it with the product decomposition, $\mathcal{E}'$.  The computation from Proposition \ref{prop:alg01} shows that the morphism $F_{\mathit{cell}}(\Sigma)$ is the immersed homotopy class of the immersed DGA map
\[
D = \big(\mathcal{A}(\Lambda, \mathcal{E}) \stackrel{i_1}{\rightarrow} (\alg_0 * \widehat{\alg_0} * \alg_1, \partial_\Gamma) \stackrel{i_0}{\hookleftarrow} \mathcal{A}(\Lambda, \mathcal{E}) \big)
\]
where $\alg_0$ and $\alg_1$ are two copies of $\mathcal{A}(\Lambda, \mathcal{E})$ and $(\alg_0 * \widehat{\alg_0} * \alg_1, \partial_\Gamma)$ is the standard mapping cylinder DGA for the identity map on $\mathcal{A}(\Lambda,\mathcal{E})$.  On the other hand, the identity morphism in $\mathfrak{DGA}^\rho_{\mathit{im}}$ is just $[I]$ where $I$ is the immersed DGA map
\[
I = \big(\mathcal{A}(\Lambda, \mathcal{E})  \stackrel{\mathit{id}}{\rightarrow} \mathcal{A}(\Lambda, \mathcal{E}) \stackrel{\mathit{id}}{\leftarrow}  \mathcal{A}(\Lambda, \mathcal{E})\big).
\]
Note that the immersed map $D$ is obtained from two copies of $I$ precisely as in the statement of Proposition \ref{prop:CompositionAlt}, which therefore shows that $D$ is immersed homotopic to $I \circ I = I$ as required.

\end{proof}

\subsection{Isomorphism between the cellular and immersed LCH functors}  \label{sec:4-5} In this section,
we state  a precise relationship between the immersed and cellular LCH functors, $F$ and $F_{\comp}$, with the proof as well as some technical details postponed until the Appendices \ref{app:iso} and \ref{app:A}.

The functors $F:\leg_{\mathit{im}}^\rho\to \dga_{im}^\rho$ and $F_{\mathit{cell}}:\mathfrak{Leg}^\rho_{\mathit{cell}} \rightarrow \mathfrak{DGA}^\rho_{\mathit{im}}$  have slightly different domain categories.  In $\leg_{im}^\rho$ (resp. $\mathfrak{Leg}^\rho_{\mathit{cell}}$), the objects are (closed, $1$-dimensional) $\rho$-graded Legendrians in $J^1M$
equipped with a suitable choice of metric (resp. polygonal decomposition), while morphisms are equivalence classes of conical (resp. compact) $\rho$-graded Legendrian cobordisms.  For comparing the two functors, it is convenient to work with a third domain category, that again consists of Legendrians equipped with some additional data.

\begin{definition}
With $\rho \in \Z_{\geq 0}$ and $1$-dimensional $M$ fixed, define the {\bf transverse Legendrian category}, $\mathfrak{Leg}^\rho_{\pitchfork}$, to have objects $(\Lambda, \mathcal{E}_\pitchfork, \mu)$ where $\Lambda \subset J^1M$ is Legendrian, $\mu$ is a $\Z/\rho$-valued Maslov potential, and $\mathcal{E}_\pitchfork$ is an {\bf admissible transverse decomposition} for $\Lambda$ as defined in Appendix \ref{app:iso}, Definition \ref{def:1dimDec}.  Morphisms are compact $\rho$-graded Legendrian cobordisms up to combinatorial equivalence of front and base projections (as in Section \ref{sec:4-4}).
\end{definition}

An admissible transverse decomposition of $\Lambda$ is a cellular decomposition of a neighborhood of the base projection $\pi_x(\Lambda) \subset M$, having cells transverse to the projection of the singular set of $\Lambda$,
subject to some technical restrictions that will not be relevant in the main body of the article.  At this point, it is sufficient to note that any $1$-dimensional Legendrian $\Lambda \subset J^1M$ with generic front and base projections admits an admissible transverse decomposition.

In Appendix \ref{app:iso}, two functors are constructed
\[
\begin{array}{ccc}
G_\mathit{std}: \mathfrak{Leg}^\rho_{\pitchfork} \rightarrow \mathfrak{Leg}^\rho_{\mathit{im}},  & \quad \quad \mbox{and} \quad \quad & G_{||}: \mathfrak{Leg}^\rho_{\pitchfork} \rightarrow \mathfrak{Leg}^\rho_{\comp},  \\
(\Lambda, \mathcal{E}_\pitchfork) \mapsto (\widetilde{\Lambda}, \widetilde{g}) & &  (\Lambda, \mathcal{E}_\pitchfork) \mapsto (\Lambda, \mathcal{E}_{||})
\end{array}
\]
where $\widetilde{\Lambda}$, called the {\bf standard geometric model} for $\Lambda$, is Legendrian isotopic to $\Lambda$ with front and base projections combinatorially equivalent to those of $\Lambda$. The actions on morphisms are as follows:
\begin{itemize}
\item The morphism spaces in $\mathfrak{Leg}^\rho_{\pitchfork}$ and $\mathfrak{Leg}^\rho_{\comp}$ only depend on $\Lambda$ and are the same in both categories, and $G_{||}$ acts as the identity on morphisms.
\item Given a compact cobordism $\Sigma$ from $\Lambda_0$ to $\Lambda_1$, we can choose any conical Legendrian cobordism $\Sigma' \subset J^1(\R_{>0} \times M)$ from $G_{\mathit{std}}(\Lambda_0)$ to $G_{\mathit{std}}(\Lambda_1)$ whose front and base projections are  combinatorially equivalent to $\Sigma$ (after truncating the conical ends).  The conical Legendrian isotopy type of $\Sigma'$ depends only on $\Sigma$, and we have $G_{\mathit{std}}([\Sigma]) = [ \Sigma']$.
\end{itemize}

The standard geometric model, $G_\mathit{std}(\Lambda, \mathcal{E}_\pitchfork)= (\widetilde{\Lambda}, \widetilde{g})$, is constructed so that the Reeb chords of $\widetilde{\Lambda}$ closely match the generators of the cellular DGA, $\mathcal{A}(\Lambda, \mathcal{E}_{||})$.  In fact, $\mathcal{A}(\widetilde{\Lambda}, \widetilde{g})$ is isomorphic to a stabilization of $\mathcal{A}(\Lambda, \mathcal{E}_{||})$, and a particular homotopy equivalence
\[
q_{\Lambda}: \mathcal{A}(\widetilde{\Lambda}, \widetilde{g}) \rightarrow \mathcal{A}(\Lambda, \mathcal{E}_{||})
\]
that we call the {\bf canonical quotient map} is identified in Section \ref{sec:Iso1dim}.  As $q_\Lambda$ is invertible in the DGA homotopy category $\mathfrak{DGA}^\rho_{\mathit{hom}}$, it follows that the immersed map
\[
Q_{\Lambda} = \big[\mathcal{A}(\widetilde{\Lambda}, \widetilde{g}) \stackrel{q_\Lambda}{\rightarrow} \mathcal{A}(\Lambda, \mathcal{E}_{||}) \stackrel{\mathit{id}}{\hookleftarrow} \mathcal{A}(\Lambda, \mathcal{E}_{||}) \big]  \in \dga^\rho_{\mathit{im}}
\]
(that is the image of $q_\Lambda$ under the functor from Proposition \ref{prop:IFunctor}) is an isomorphism in $\dga^\rho_{\mathit{im}}$.

\begin{proposition}  \label{prop:Fisomorph}
The canonical quotient map construction,
\[
(\Lambda, \mathcal{E}_\pitchfork) \in \mathfrak{Leg}^\rho_\pitchfork   \, \mapsto \, Q_{\Lambda} \in Hom_{\dga^\rho_{\mathit{im}}}(\mathcal{A}(\widetilde{\Lambda}, \widetilde{g}), \mathcal{A}(\Lambda, \mathcal{E}_{||}))
\]
gives an isomorphism (invertible natural transformation) from the functor that is the composition
\[
\mathfrak{Leg}^\rho_{\pitchfork} \stackrel{G_{std}}{\rightarrow} \mathfrak{Leg}^\rho_{im} \stackrel{F}{\rightarrow} \dga^\rho_{\mathit{im}}.
\]
to the functor
\[
 \mathfrak{Leg}^\rho_{\pitchfork} \stackrel{G_{||}}{\rightarrow} \mathfrak{Leg}^\rho_{\comp} \stackrel{F_{\mathit{cell}}}{\rightarrow} \dga^\rho_{\mathit{im}}.
\]
That is, the $Q_{\Lambda}$ are invertible, and for any compact Legendrian cobordism $\Sigma: (\Lambda_-, \mathcal{E}^{\pitchfork}_-) \rightarrow (\Lambda_+, \mathcal{E}^{\pitchfork}_+)$ with generic base and front projection
we have a commutative diagram in $\dga^\rho_{\mathit{im}}$,
\begin{equation} \label{eq:QLambda}
\xymatrix{ \mathcal{A}(\widetilde{\Lambda}_+, \widetilde{g}_+)\ar[rr]^{F\circ G_{\mathit{std}}(\Sigma)} \ar[d]_{Q_{\Lambda_+}} & & \mathcal{A}(\widetilde{\Lambda}_-, \widetilde{g}_-) \ar[d]^{Q_{\Lambda_-}}  \\ \calg(\Lambda_+, \mathcal{E}^{||}_+) \ar[rr]_{F_{\mathit{cell}}\circ G_{||}(\Sigma)}  & & \calg(\Lambda_-, \mathcal{E}^{||}_-) }.
\end{equation}
\end{proposition}

In summary, the following diagram is commutative up to a canonical isomorphism of functors:
\[
\xymatrix{ & \mathfrak{Leg}^\rho_{im}  \ar[rd]^{F} & \\ \mathfrak{Leg}^\rho_{\pitchfork} \ar[ru]^{G_{\mathit{std}}}  \ar[rd]^{G_{||}}  & & \dga^\rho_{\mathit{im}} \\ & \mathfrak{Leg}^\rho_{\comp} \ar[ru]^{F_{\mathit{cell}}} & }
\]

\begin{remark}
An arbitrary Legendrian $\Lambda \subset J^1M$
 is related to one
in the image of $G_{std}$ by a Legendrian isotopy (that preserves the combinatorial appearance of the front projection).  Thus Proposition \ref{prop:Fisomorph} allows the cellular DGA to be used for computing immersed maps $F(\Sigma)$ induced by a conical Legendrian cobordism $\Sigma:\Lambda_-\rightarrow \Lambda_+$ after possibly changing the Legendrians at the ends from $\Lambda_\pm$ to $\Lambda'_{\pm}$ by Legendrian isotopy.  Composing $\Sigma$ with cobordisms $\Sigma_+:\Lambda_+ \rightarrow \Lambda_+'$ and $\Sigma_-: \Lambda_-'\rightarrow \Lambda_-$ induced by the isotopies modifies $\Sigma$ to a cobordism $\Sigma': \Lambda_-' \rightarrow \Lambda_+'$ to which the cellular computation of Proposition \ref{prop:Fisomorph} can be applied directly.
 Moreover, if one wants to work with the original  $\Lambda_\pm$, then the maps $F(\Sigma_-)$ and $F(\Sigma_+)$ can be
explicitly computed as in \cite[Section 6]{EHK}.
\end{remark}

The following corollary of Proposition \ref{prop:Fisomorph} shows that the immersed DGA maps associated to a compact cobordism $\Sigma$ by the cellular functor, $F_{\comp}$, only depend on the Legendrian isotopy type of $\Sigma$.

\begin{corollary} \label{cor:LegInv}
Given $(\Lambda_0, \mathcal{E}_0), (\Lambda_1, \mathcal{E}_1) \in \mathfrak{Leg}^\rho_{\comp}$ and compact cobordisms $\Sigma$ and $\Sigma'$.  If $\Sigma$ and $\Sigma'$ are Legendrian isotopic rel. boundary, then $F_\comp(\Sigma) = F_\comp(\Sigma')$ in $\mathfrak{DGA}^\rho_{\mathit{im}}$.
\end{corollary}

\begin{proof}

\noindent {\bf Case 1.}  Suppose $(\Lambda_k,\mathcal{E}_k)= G_{||}(\Lambda_k,\mathcal{E}_k^\pitchfork)$  for some admissible transverse decompositions $\mathcal{E}_k^\pitchfork$ for $\Lambda_k$, $k=0,1$.

\medskip

In this case, Proposition \ref{prop:Fisomorph} allows us to compute in $\mathfrak{DGA}^\rho_{\mathit{im}}$,
\[
F_{\comp}(\Sigma) = Q_{\Lambda_1} \circ F( G_{\mathit{std}}(\Sigma)) \circ Q_{\Lambda_0}^{-1} = Q_{\Lambda_1} \circ F( G_{\mathit{std}}(\Sigma')) \circ Q_{\Lambda_0}^{-1} = F_{\comp}(\Sigma').
\]
[At the second equality we used that $F(G_{\mathit{std}}(\Sigma))$ only depends on the conical Legendrian isotopy type of $G_{\mathit{std}}(\Sigma)$.]

\medskip

\noindent {\bf Case 2.}  For general $\mathcal{E}_0$ and $\mathcal{E}_1$.

\medskip

In this case, fix some choice of admissible transverse decompositions $\mathcal{E}_k^\pitchfork$ for $\Lambda_k$, $k=0,1$.  Let $I_k$ denote the product cobordism $j^1(1_{[0,1]} \cdot \Lambda_k)$ viewed as a morphism in $\mathit{Hom}_{\mathfrak{Leg}^\rho_{\comp}}( (\Lambda_k, \mathcal{E}_k), (\Lambda_k, G_{||}(\mathcal{E}_k^\pitchfork))$, and let $\Sigma_\pitchfork$ and $\Sigma_\pitchfork'$ denote $\Sigma$ and $\Sigma'$ viewed as morphisms in $\mathit{Hom}_{\mathfrak{Leg}^\rho_{\comp}}( (\Lambda_0, G_{||}(\mathcal{E}_0^\pitchfork)), (\Lambda_1, G_{||}(\mathcal{E}_1^\pitchfork))$.  Using Case 1, we compute
\begin{align*}
F_{\comp}(\Sigma) = F_{\comp}(I_1^{-1} \circ \Sigma_\pitchfork \circ I_0) &= F_{\comp}(I_1^{-1}) \circ F_{\comp}(\Sigma_\pitchfork) \circ F_{\comp}(I_0) \\
 &= F_{\comp}(I_1^{-1}) \circ F_{\comp}(\Sigma_\pitchfork') \circ F_{\comp}(I_0) = F_{\comp}(\Sigma').
\end{align*}

\end{proof}

As in Section \ref{sec:augimmersed}, given an immersed DGA map $M= \left(\mathcal{A}_1 \stackrel{f}{\rightarrow} \mathcal{B} \stackrel{i}{\hookleftarrow} \mathcal{A}_2 \right)$ we write $I(M) = \mathit{Im}(i^*\times f^*) \subset \mathit{Aug}_\rho(\mathcal{A}_1)/{\sim} \times \mathit{Aug}_\rho(\mathcal{A}_2)/{\sim}$ for the {\it induced augmentation set} of $M$. According to Proposition \ref{prop:indaug}, $I(M)$ only depends on the immersed homotopy class of $M$.

\begin{corollary}  \label{cor:augmentationset}  Let $\Sigma \subset J^1([0,1]\times M)$ be a compact Legendrian cobordism from $\Lambda_0$ to $\Lambda_1$.  Let $\mathcal{E}_k^\pitchfork$ be an admissible transverse decomposition for $\Lambda_k$, $k=0,1$, and write $G_{std}(\Lambda_k,\mathcal{E}_k^\pitchfork) = (\widetilde{\Lambda}_k, \widetilde{g}_k)$ and $G_{||}(\Lambda_k,\mathcal{E}_k^\pitchfork) =(\Lambda_k,\mathcal{E}_k^{||})$.
Then, the induced augmentation sets
$I_\Sigma := I(F(G_{\mathit{std}}(\Sigma)))$ and $I^{\mathit{cell}}_{\Sigma} := I(F_\mathit{cell}(G_{||}(\Sigma)))$ are related by
\begin{equation} \label{eq:qlambda}
(q_{\Lambda_0}^* \times q_{\Lambda_1}^*)\big(I^{\mathit{cell}}_{\Sigma} \big) = I_\Sigma
\end{equation}
where
\[
q_{\Lambda_0}^* \times q_{\Lambda_1}^*:\aug_\rho(\Lambda_0, \mathcal{E}_0^{||})/{\sim} \times \aug_\rho(\Lambda_1, \mathcal{E}_1^{||})/{\sim} \stackrel{\cong}{\rightarrow} \aug_\rho(\widetilde{\Lambda}_0, \widetilde{g}_0)/{\sim} \times \aug_\rho(\widetilde{\Lambda}_1, \widetilde{g}_1)/{\sim}
\]
 is the bijection induced by the canonical quotient maps,  $q_{\Lambda_k}: \alg(\widetilde{\Lambda}_k, \widetilde{g}_k) \rightarrow \alg(\Lambda_k, \mathcal{E}_k^{||})$.
\end{corollary}

\begin{proof}
  Applying the functor from Proposition \ref{prop:relfunctor} to (\ref{eq:QLambda}) produces the diagram of relations
\begin{equation} \label{eq:QLambdaRel}
\xymatrix{ \aug(\widetilde{\Lambda}_1), \widetilde{g}_1)/{\sim}  & \aug(\widetilde{\Lambda}_0, \widetilde{g}_0)/{\sim} \ar[l]_{I_\Sigma}  \\ \aug(\Lambda_1, \mathcal{E}^{||}_1)/{\sim} \ar[u]^{I(Q_{\Lambda_1})}  & \aug(\Lambda_0, \mathcal{E}_0)/{\sim} \ar[l]_{I^{\mathit{cell}}_\Sigma} \ar[u]_{I(Q_{\Lambda_0})} }.
\end{equation}
As noted in \ref{sec:embeddedcase}, the induced augmentation sets associated to $Q_{\Lambda_{k}}$ are just the graphs of $q^*_{\Lambda_{k}}$, so Observation \ref{obs:relation} (2) and (3) can be applied to arrive at (\ref{eq:qlambda}).
\end{proof}

config: arxiv


TITLE: Is there a McDiarmid-type inequality for sequences with a finite range of dependence?
QUESTION [6 upvotes]: Let $X, X_1, X_2, \ldots, X_N$ be a sequence of identically distributed random variables with $0 \leq X \leq 1$.
We do not assume the sequence is iid, but rather allow the random variables to be dependent on their neighbors: we suppose that there exists $k\geq 1$ such that for every $I,J \subseteq \{ 1,\ldots, N\}$ satisfying $\min\{ |i-j| \,:\, i\in I,\, j\in J \} \geq k$, the collections $\{ X_i\,:\, i\in I\}$ and $\{ X_j \,:\, j\in J \}$ are independent. (It might help to just consider the case $k=2$.)
Let $f: \mathbb{R}^N \to \mathbb R$ be a Lipschitz function with Lipschitz constant one.
Here is my question: is there a good concentration inequality for $f(X_1,\ldots,X_N)$?
Let me recall two well-known facts:

If $k=1$, then the sequence is i.i.d., and we have McDiarmid's inequality, which states that:
\begin{equation}
\mathbb{P} \left[ \left| f(X_1,\ldots,X_N) - \mathbb E \left[ f(X_1,\ldots,X_N) \right] \right| > \lambda  \right] \leq \exp \left( -\frac{2\lambda^2}{N} \right).
\end{equation}
On the other hand, if $f(X_1,\ldots,X_N) = X_1 + \cdots+X_N$, then I can run the proof of Bernstein's inequalities (the usual exponential generating function argument) combined with a neat Holder inequality trick to prove a similar bound.

However, I am interested in general, non-affine, Lipschitz $f$, and the proof of McDiarmid's inequality uses a martingale argument that doesn't seem to work very well with my weaker independence assumption (the Holder trick doesn't seem to me to work). I have tried googling without success. Does a concentration inequality like McDiarmid's inequality survive? I would even be happy with something not quite as strong.

REPLY [8 votes]: There are several versions of this type:
1) K. Marton has results for dependent variables. Maybe closest to what you ask (for convex functions $f$) is the paper of Samson:
Samson paper
2) For a martingale difference based argument, see Kontorovich and Ramanan:
Kontorovich-Ramanan paper
The references of these articles give more pointers...


TITLE: The rank of a matrix, dependent on the value of $t$
QUESTION [2 upvotes]: I'm trying to analyse the rank of the following matrix, for $t\in \mathbb{R}$.
$$\begin{bmatrix}
    t+3 & 5 & 6 \\
    -1 & t-3 & -6 \\
    1 & 1 & t+4
\end{bmatrix}$$
With $R_1\leftrightarrow R_3$, $-(t+3)R_1+R_3 \rightarrow R_3$, $R_1+R_2 \rightarrow R_2$, and finally $-R_2+R_3\rightarrow R_3$. I get
$$\begin{bmatrix}
    1 & 1 & t-4 \\
    0 & t-2 & t-2 \\
    0 & 0 & -t^2-8t-4
\end{bmatrix}$$
And this makes that when $t=2$ or $t=2(-2 \pm \sqrt{3})$. However, these values are not what they are supposed to give(-2,2,4). Where did I make a mistake? Any help would be appreciated.

REPLY [1 votes]: \begin{align}
\begin{bmatrix}
    t+3 & 5 & 6 \\
    -1 & t-3 & -6 \\
    1 & 1 & t+4
\end{bmatrix}
&\to
\begin{bmatrix}
    1 & 1 & t+4 \\
    -1 & t-3 & -6 \\
    t+3 & 5 & 6 \\
\end{bmatrix}
&&R_1\leftrightarrow R_3
\\
&\to
\begin{bmatrix}
1   & 1   & t+4 \\
0   & t-2 & t-2 \\
t+3 & 5   & 6 \\
\end{bmatrix}
&&R_2\gets R_2+R_1
\\
&\to
\begin{bmatrix}
1 & 1   & t+4 \\
0 & t-2 & t-2 \\
0 & 2-t & -t^2-7t-6 \\
\end{bmatrix}
&&R_3\gets R_3-(t+3)R_1
\\
&\to
\begin{bmatrix}
1 & 1   & t+4 \\
0 & t-2 & t-2 \\
0 & 0   & -t^2-6t-8 \\
\end{bmatrix}
&&R_3\gets R_3+R_1
\end{align}
The roots of $-t^2-6t-8$ are $-2$ and $-4$.

If $t\notin\{2,-2,-4\}$ the rank is $3$.
If $t=2$, the rank is $2$.
If $t=-2$ or $t=-4$, the rank is $2$.

The result you're given is wrong. The numbers $-2$, $2$ and $4$ are the eigenvalues of
$$
A=\begin{bmatrix}
3 & 5 & 6 \\
-1 & -3 & -6 \\
1 & 1 & 4
\end{bmatrix}
$$
and your case is finding the eigenvalues of $-A$.


TITLE: Sums of subsets of $\mathbb{Z}/n\mathbb{Z}$
QUESTION [11 upvotes]: I have encountered a problem that I suspect has been thoroughly studied but I have not been able to find references. Can anyone point me to a published reference dealing with this or a closely related problem?
Here is the problem:
Let $A$ and $B$ be disjoint $k$-subsets of $\mathbb{Z}/n\mathbb{Z}$.  Consider $S(A,B)=\sum_{x\in A}x - \sum_{y\in B}y$. As $(A,B)$ ranges over all possible ordered pairs of disjoint $k$-subsets of $\mathbb{Z}/n\mathbb{Z}$, how are the sums $S(A,B)$ distributed over the elements of $\mathbb{Z}/n\mathbb{Z}$?  More precisely, for how many of the $\binom{n}{k}\binom{n-k}{k}$ choices of $(A,B)$ is $S(A,B)$ equal to each of the elements of $\mathbb{Z}/n\mathbb{Z}$?
Again I am just looking for references.  I actually have a solution in the case that n is prime, but I assume the result is known for more general n. I would be interested in any leads.

REPLY [6 votes]: For convenience I will first consider how many sums $S(A,B)$ are equal
to 0. I might have made some computational errors, but I think that
the method is correct. Regard $n$ as fixed. Let $f_k$ be the number of
pairs $(A,B)$ of $k$-element subsets (not necessarily disjoint) of
$\mathbb{Z}/n\mathbb{Z}$ such that $\sum_{x\in A}x-\sum_{y\in
B}y=0$. Let $g_k$ be the same, except that $A$ and $B$ must be
disjoint. It is easy to see that
  $$ f_k = g_k+{n-2(k-1)\choose 1}g_{k-1}+{n-2(k-2)\choose
    2}g_{k-2}+\cdots. $$
These equations for $0\leq k\leq n$ can be inverted to express
the $g$'s in terms of the $f$'s, so it suffices to find $f_k$.
Let $[x^iy^j]P(x,y)$ denote the coefficient of $x^iy^j$ in the
polynomial $P(x,y)$.  By standard properties of roots of unity, we have
  \begin{eqnarray*} f_k & = & [x^ky^k] \frac 1n \sum_{\zeta^n=1}
  \prod_{j=0}^{n-1} (1+x\zeta^j) (1+y\zeta^{-j})\\ & = &
   \frac 1n\sum_{d|n}\phi(d)(1-(-x)^{n/d})^d (1-(-y)^{n/d})^d.
  \end{eqnarray*}
It is routine to extract the coefficient of $x^ky^k$,
Completely analogous reasoning works for sums equal to any $m\in
\mathbb{Z}/n\mathbb{Z}$, though the computation becomes messier. A
related computation appears in
http://math.mit.edu/~rstan/pubs/pubfiles/8g.pdf. See also Enumerative
Combinatorics, vol. 1, Exercise 1.105.


TITLE: parametrization of a moving wheel
QUESTION [0 upvotes]: Attempt:
Given, radius of wheel $0.5 m$
Total distance travelled by the point during first revolution = circumference of the wheel
therefore, distance travelled by the point $2 \pi r$
where r is the radius of the wheel
distance travelled by the point $2 \cdot 3.14 \cdot 0.5 = 3.14 m$
Now, a circle can be defined as the locus of the all points on it that satisfies the equations,
$x = r cos(t), y = rsin(t)$
where x, y are the coordinates of any point on the circle, r is the radius of the circle and t is the parameter - the angle subtended by the point at the circle's center
so the parametrization of the path would be
$x = 0.5cos(t), y = 0.5sin(t)$
Is it right?

REPLY [2 votes]: That parametrization is with respect to the midpoint of the wheel. But note that the midpoint is moving as well! You'll need to account for that movement, too, see https://en.wikipedia.org/wiki/Cycloid.


TITLE: Next book in learning Differential Geometry
QUESTION [8 upvotes]: I have just finished the book "Manfredo P. do Carmo - Differential Geometry of Curves and Surfaces". My aim is to reach to graduate level  to do research, but articles are not only too advanced to study after Carmo's book, but also I don't think that they are readable by just studying Carmo's book at all for a self-learner like me. Please someone tell me a book for Differential Geometry more advanced than Carmo's book but readable esp. for self-learning . Thanks a lot.

REPLY [8 votes]: Here's my answer to this question at length.
http://www.geometry.org/tex/conc/differential_geometry_books.html
In summary, if you are looking at the pure mathematics style of DG, you would be looking at do Carmo's "Riemmanian geometry", three books by John M. Lee, and for something more advanced, the books by Serge Lang, Bishop and Crittenden and Peter Petersen. And for the really advanced level, there's Schoen and Yau "Lectures on Differential Geometry", which lists many hundreds of open problems to work on at the postgraduate level.
There are also a few items on this web site which address the same question, some of them several years ago.