As a first step, we construct the hypercube shown in Equation (<ref>).

\begin{equation}
\begin{tikzcd}[
column sep={2.3cm,between origins},
row sep=1.4cm,
fill opacity=.7,
text opacity=1,
\bF[U]	\ar[dr, "\id", pos=.6]
\ar[ddd, "\Theta_{\bar \b', \bar \b}"]
\ar[rr, "\id"]
\ar[ddddrrr,dotted, "\omega_{\bar \b'', \bar \b'}"]
&[.7 cm]
\ar[rd, "\id"]
\ar[ddd, "\Theta_{\bar \b', \bar \b'}"]
&[.7 cm]
\,
\\
&[.7 cm]
\bF[U]
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr,  "\id"]
&[.7 cm]
\bF[U]
\ar[ddd, "\Theta_{\bar \b'', \bar \b'}"]
\\
\\
\CF(\bar \bs', \bar \bs)
\ar[dr,"f_{\bar \b'\to \bar \b''}^{\bar \b}"]
\ar[rr, "f_{\bar \b'}^{\bar \b\to\bar \b'}"]
\ar[drrr,dashed, "h_{\bar \b'\to \bar \b''}^{\bar \b \to \bar \b'}"]
&[.7 cm]\,&
\CF(\bar \bs', \bar \bs')
\ar[dr, "f_{\bar \b'\to \bar \b''}^{\bar \b'}"]
&[.7 cm]
\\
& [.7 cm]
\CF(\bar \bs'', \bar \bs)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu, "\Theta_{\bar \b'', \bar \b}"]
\ar[rr, "f_{\bar \b''}^{\bar \b\to \bar \b'}"]
\ar[from=uuuul,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuul, dashed, "\lambda_{\bar \b'', \bar \b}"]
&[.7 cm]
\CF(\bar \bs'', \bar \bs')
\ar[from=uuull,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuull,dashed,"\lambda_{\bar \b'', \bar \b'}"]
\end{tikzcd}
\label{eq:top-degree-generator-hypercube}
\end{equation}

The hypercube in Equation (<ref>) is constructed similarly to the procedure for filling a hypercube of handleslide equivalent attaching curves, described by Manolescu and Ozsváth <cit.>*Lemma 8.6. In Equation (<ref>), an arrow labeled by an element $\xs\in \bT_{\a}\cap \bT_{\b}$ means the map from $\bF[U]$ to $\CF(\as,\bs)$ given by

\[
1\mapsto \xs,
\]

extended equivariantly over $U$. The existence of $\lambda_{\bar \b'',\bar \b'}$, $\lambda_{\bar \b'', \bar \b'}$ and $\omega_{\bar \b'', \bar \b'}$ so that the diagram in (<ref>) satisfies the hypercube relations follows from the fact that $\bar \bs$, $\bar\bs'$ and $\bar\bs''$ are handleslide equivalent.

It is helpful to concretely understand the hypercube relations in (<ref>). The length 1 relations amount to each chain marked with $\Theta$ being a cycle. The length 2 relations are as follows:

\begin{equation}
\begin{split}
f_{\bar \b'}^{ \bar \b\to \bar \b'}(\Theta_{\bar \b', \bar \b})&=\Theta_{\bar \b', \bar \b'}\\
f_{\bar \b'\to \bar \b''}^{\bar \b'}(\Theta_{\bar \b', \bar \b'})&=\Theta_{\bar \b'', \bar \b'}\\
\Theta_{\bar \b'', \bar \b}+f_{\bar \b'\to \bar \b''}^{\bar \b}(\Theta_{\bar \b', \bar \b})&=\d \lambda_{\bar \b'', \bar \b}\\
f_{\bar \b''}^{\bar \b\to \bar \b'}(\Theta_{\bar \b'', \bar \b})+ \Theta_{\bar \b'', \bar \b'}&= \d \lambda_{\bar \b'', \bar \b'}\\
f_{\bar \b'\to \bar \b''}^{\bar \b'}\circ f_{\bar \b'}^{\bar \b\to \bar \b'}+f_{\bar \b''}^{\bar \b\to \bar \b'}\circ f_{\bar \b'\to \bar \b''}^{\bar \b}&=\left[\d, h_{\bar \b'\to \bar \b''}^{\bar \b\to \bar \b'}\right].
\end{split}
\label{eq:hypercube-relations-diagonal-product}
\end{equation}

Note that the first relation is forced by grading considerations, and the second is implied by a nearest point map computation. The third and fourth relations are clearly satisfiable, and the final relation is the associativity relations for quadrilaterals.

Finally, the length 3 relation is

\[
f_{\bar \b''}^{\bar \b \to \bar \b'}(\lambda_{\bar \b'', \bar \b})+\lambda_{\bar \b'', \bar \b'}+h_{\bar \b'\to \bar \b''}^{\bar \b \to \bar \b'}(\Theta_{\bar \b', \bar \b})=\d \omega_{\bar \b'', \bar \b'}.
\]

Note that the hypercube in (<ref>) is a new type of hypercube. It is not naturally realized by pairing alpha and beta hypercubes together. We can think of the left side as determining the following hypercube of alpha attaching curves

\begin{equation}
\begin{tikzcd}[labels=description, column sep=1.5cm, row sep=1.5cm]
\bar{\bs}
\ar[r,"1"]
\ar[dr,dashed,"\lambda_{\bar\b'',\bar\b}"]
\ar[d,"\Theta_{\bar \b', \bar \b}"]
\ar[d, "\Theta_{\bar\b'',\bar\b}"]
\\
\bar{\bs}'
\ar[r, "\Theta_{\bar\b'',\bar\b'}"]
\end{tikzcd}
\label{eq:subface-left-pure-1-handle}
\end{equation}

On the other hand, the front side is more naturally viewed as corresponding to the following hypercube of beta attaching curves

\begin{equation}
\begin{tikzcd}[labels=description, column sep=1.5cm, row sep=1.5cm]
\bar{\bs}''
\ar[r,"1"]
\ar[dr,dashed,"\lambda_{\bar\b'',\bar\b'}"]
\ar[d,"\Theta_{\bar \b'', \bar \b}"]
\ar[d, "\Theta_{\bar\b'',\bar\b'}"]
\\
\bar{\bs}
\ar[r, "\Theta_{\bar\b,\bar\b'}"]
\end{tikzcd}
\label{eq:subface-front-pure-1-handle}
\end{equation}

The bottom face of (<ref>) is the pairing of hypercubes of attaching curves.

Next, we pick hypercubes of alpha attaching curves and beta attaching curves (respectively):

\begin{equation}
\begin{tikzcd}[labels=description, column sep=1.5cm, row sep=1.5cm]
\as
\ar[r,"\Theta_{\a',\a}"]
\ar[dr,dashed,"\lambda_{\a'',\a}"]
\ar[d,"1"]
\ar[d, "\Theta_{\a'',\a'}"]
\\
\as
\ar[r, "\Theta_{\a'',\a}"]
\end{tikzcd}
\quad \text{and} \quad
\begin{tikzcd}[labels=description, column sep=1.5cm, row sep=1.5cm]
\bs
\ar[r,"\Theta_{\b,\b'}"]
\ar[dr,dashed,"\lambda_{\b,\b''}"]
\ar[d,"1"]
\ar[d, "\Theta_{\b',\b''}"]
\\
\bs
\ar[r, "\Theta_{\b,\b''}"]
\end{tikzcd}
\label{eq:alpha-hypercubes}
\end{equation}

As usual, we assume that the length 1 morphisms represent top degree cycles. Note that we do not assume that the chains in (<ref>) have any relation with the chains in (<ref>) or (<ref>).

\[
\begin{tikzcd}[
column sep={2.1cm,between origins},
row sep=1.5cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;
\ar[dr, "\id"]
\ar[ddd, "F_1^{c \bar{\b}, c \bar{\b}}"]
\ar[rr, "f_{\a\to \a'}^{\b}"]
\ar[ddddrrr,dotted]
&[.7 cm]
\ar[rd, "f_{\a'\to \a''}^{\b}"]
\ar[ddd, "F_1^{c\bar{\b}', c \bar{\b}}"]
\ar[rr, "f_{\a'}^{\b\to \b'}"]
\ar[ddddrrr,dotted]
\ar[ddddr,dashed]
&[.7 cm]
\ar[dr, "f_{\a'\to \a''}^{\b'}"]
\ar[ddd, "F_{1}^{c \bar{\b}', c \bar{\b}'}"]
&[.7 cm]
\\
&[.7 cm]
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "f_{\a\to \a''}^{\b}"]
&[.7 cm]
\ar[ddd, "F_1^{c\bar{\b}'', c\bar{\b}}"]
\ar[rr, "f_{\a''}^{\b\to \b'}"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll, dashed]
\ar[ddd, "F_{1}^{c\bar{\b}'', c\bar{\b}'}"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll, dashed]
\\
\\
(\as c \bar{\bs}, \bs c \bar{\bs})
\ar[dr,"\id"]
\ar[rr, "f_{\a c \bar{\b}\to \a' c \bar{\b}'}^{\b c \bar{\b}}"]
\ar[drrr,dashed]
&[.7 cm]\,&
(\as' c \bar{\bs}', \bs c \bar{\bs})
\ar[dr, "f_{\a' c \bar{\b}'\to \a'' c \bar{\b}''}^{\b c \bar{\b}}"]
\ar[rr, "f_{\a'c \bar{\b}'}^{\b c \bar{\b}\to \b' c \bar{\b}'}"]
\ar[drrr,dashed]
&[.7 cm]\,
(\as'c\bar{\bs}',\bs' c \bar{\bs}')
\ar[dr, "f_{\a'c \bar{\b}'\to \a'' c \bar{\b}''}^{\b' c \bar{\b}'}"]
\\
& [.7 cm]
|[alias=D]|(\as c \bar{\bs}, \bs c \bar{\bs})
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu, "F_1^{c\bar{\b}, c\bar{\b}}"]
\ar[rr, "f_{\a c \bar{\b}\to \a'' c \bar{\b}''}^{\b c \bar{\b}}"]
&[.7 cm]
(\as'' c \bar{\bs}'', \bs c \bar{\bs})
\ar[rr, "f_{\a''c \bar{\b}''}^{\b c \bar{\b}\to \b' c \bar{\b}'}" ]
|[alias=C]|(\as'' c \bar{\bs}'', \bs' c \bar{\bs}')
\ar[from=uuull,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuull,dashed]
\end{tikzcd}
\]

\[
\begin{tikzcd}[
column sep={2.1cm,between origins},
row sep=1.5cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;
\ar[dr, "\id"]
\ar[ddd, "F_1^{c \bar{\b}, c \bar{\b}}"]
\ar[rr, "f_{\a\to \a''}^{\b}"]
&[.7 cm]
\ar[rd, "\id"]
\ar[ddd, "F_1^{c\bar{\b}'', c \bar{\b}}"]
\ar[rr, "f_{\a''}^{\b\to \b'}"]
\ar[ddddrrr,dotted]
\ar[dddrr,dashed]
&[.7 cm]
\ar[dr, "f_{\a''}^{\b'\to \b''}"]
\ar[ddd, "F_{1}^{c \bar{\b}'', c \bar{\b}'}"]
&[.7 cm]
\\
&[.7 cm]
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "f_{\a\to \a''}^{\b}"]
&[.7 cm]
\ar[ddd, "F_1^{c\bar{\b}'', c\bar{\b}}"]
\ar[rr, "f_{\a''}^{\b\to \b''}"]
\ar[ddd, "F_{1}^{c\bar{\b}'', c\bar{\b}''}"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll, dashed]
\\
\\
|[alias=D]|(\as c \bar{\bs}, \bs c \bar{\bs})
\ar[dr,"\id"]
\ar[rr, "f_{\a c \bar{\b}\to \a'' c \bar{\b}''}^{\b c \bar{\b}}"]
&[.7 cm]\,&
(\as'' c \bar{\bs}'', \bs c \bar{\bs})
\ar[dr, "\id"]
\ar[rr, "f_{\a''c \bar{\b}''}^{\b c \bar{\b}\to \b' c \bar{\b}'}"]
\ar[drrr,dashed]
&[.7 cm]\,
|[alias=C]|(\as''c\bar{\bs}'',\bs' c \bar{\bs}')
\ar[dr, "f_{\a''c \bar{\b}''}^{\b' c \bar{\b}'\to \b'' c \bar{\b}''}"]
\\
& [.7 cm]
(\as c \bar{\bs}, \bs c \bar{\bs})
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu, "F_1^{c \bar{\b}, c \bar{\b}}"]
\ar[rr, "f_{\a c \bar{\b}\to \a'' c \bar{\b}''}^{\b c \bar{\b}}"]
&[.7 cm]
(\as'' c \bar{\bs} '', \bs c \bar{\bs})
\ar[rr, "f_{\a''c \bar{\b}''}^{\b c \bar{\b}\to \b'' c \bar{\b}''}" ]
(\as'' c \bar{\bs}'', \bs'' c \bar{\bs}'')
\end{tikzcd}
\]

Four hypercubes. The length 2 maps are sums of quadrilateral and triangle maps (obtained by pairing hypercubes). The two boxes are stacked along the gray faces.

We now consider the diagram in Figure <ref>. We will presently define the maps appearing therein, and then subsequently prove the hypercube relations. All the maps are built from the hypercubes in (<ref>) and (<ref>). We note that the hypercube relations are slightly subtle, since they do not arise from simply pairing hypercubes of attaching curves together.

We focus first on the top left cube of Figure <ref>. The maps appearing there are as follows:

* The length 2 map along the top is the one associated to pairing the 0-dimensional beta hypercube $\bs$ with the alpha hypercube

\[
\begin{tikzcd}[labels=description, column sep=1.5cm, row sep=1.5cm]
\as
\ar[r,"\Theta_{\a',\a}"]
\ar[dr,dashed,"\lambda_{\a'',\a}"]
\ar[d,"1"]
\ar[d, "\Theta_{\a'',\a'}"]
\\
\as
\ar[r, "\Theta_{\a'',\a}"]
\end{tikzcd}
\]

* The length 2 map along the bottom is the one obtained by pairing the 0-dimensional beta hypercube $\bs c \bar \bs$ with the hypercube

\[
\begin{tikzcd}[labels=description, column sep=3cm, row sep=1.5cm]
\as c \bar \bs
\ar[r,"\Theta_{\a',\a}|\theta^+|\Theta_{\bar \b', \bar\b}"]
\ar[d,"1"]
\ar[dr,dashed, "\lambda"]
&\as' c \bar \bs'
\ar[d, "\Theta_{\a'',\a'}|\theta^+|\Theta_{\bar \b '', \bar \b '}"]
\\
\as c \bar \bs
\ar[r, "\Theta_{\a'',\a}|\theta^+|\Theta_{\bar \b'', \bar \b}"]
&\as'' c \bar \bs''
\end{tikzcd}
\]

\[
\lambda=\lambda_{\a'',\a}|\theta^+|f_{\bar{\b}'', \bar{\b}', \bar{\b}}(\Theta_{\bar{\b}'', \bar{\b}'}, \Theta_{\bar{\b}', \bar{\b}}) +\Theta_{\a'',\a}|\theta^+|\lambda_{\bar{\b}'', \bar{\b}}.
\]

* The length 2 map along the right side is

\[
\xs\mapsto f_{\a'\to \a''}^{\b}(\xs)| \theta^+| \lambda_{\bar{\b}'',\bar{\b}}.
\]

* The length 3 map is

\[
\xs \mapsto h_{\a \to \a'\to \a''}^{\b}(\xs)| \theta^+| \lambda_{\bar{\b}'', \bar \b}+f_{\a'', \a,\b}(\lambda_{\a'',\a},\xs)|\theta^+|\lambda_{\bar{\b}'',\bar{\b}}.
\]

The maps described above for the top left subcube of the diagram in Figure <ref> satisfy the hypercube relations.

The length 1 relations are immediate, as are the length 2 relations for all but the right face of the cube. For the right face of the cube, using the stabilization result from Proposition <ref>, applied to holomorphic triangles, we see that the hypercube relations amount to showing that if $\xs\in \bT_{\a''}\cap \bT_{\b}$, then

\[
f_{\a'\to \a''}^{\b}(\xs)| \theta^+|\left(\Theta_{\bar \b'', \bar \b}+f_{\bar \b'', \bar \b', \bar \b}(\Theta_{\bar \b'', \bar \b'}, \Theta_{\bar \b', \bar \b})\right)=\d \left(f_{\a' \to \a''}^{\b}(\xs)| \theta^+ | \lambda_{\bar{\b}'', \bar \b}\right)+f_{\a'\to \a''}^{\b}(\d \xs)| \theta^+|\lambda_{\bar{\b}'', \bar{\b}},
\]

which follows from the form of the differential on the Heegaard diagram obtained by attaching a 1-handle. The key point is that the right-hand side of the above equation simplifies to just $f_{\a'\to \a''}^{\b}(\xs)| \theta^+|\d \lambda_{\bar{\b}'', \bar \b}$, which coincides with the left-hand side by Equation (<ref>).

Next, we consider the length 3 relation of the cube. The desired relation is that if $\xs\in \bT_{\a}\cap \bT_{\b},$ then

\begin{equation}
\begin{split}
&h_{\a\to \a'\to \a''}^{\b}(\xs)|\theta^+|\Theta_{\bar \b'', \bar \b} \\
&+f_{\a'',\a,\b}(\lambda_{\a'',\a},\xs)|\theta^+|\Theta_{\bar \b'', \bar \b}\\
&+h_{\a'' c \bar \b'', \a' c \bar \b', \a c\bar \b, \b c \bar \b}(\Theta_{\a'',\a'}|\theta^+|\Theta_{\bar \b'' \bar \b'}, \Theta_{\a',\a}|\theta^+|\Theta_{\bar \b' \bar \b},\xs|\theta^+|\Theta_{\bar \b, \bar \b})\\
&+ (f_{\a'\to \a''}^{\b}\circ f_{\a \to \a'}^{\b})(\xs)|\theta^+|\lambda_{\bar \b'', \bar \b}\\
&+f_{\a'' c \bar \b'', \a c \bar \b, \b c \bar \b}(\lambda_{\a'',\a}|\theta^+|f_{\bar \b'', \bar \b', \bar \b}(\Theta_{\bar \b'', \bar \b'}, \Theta_{\bar \b', \bar \b}), \xs |\theta^+|\Theta^+_{\bar \b, \bar \b})\\
&+f_{\a'' c \bar \b'', \a c \bar \b, \b c \bar \b}(\Theta_{\a'', \a}|\theta^+|\lambda_{\bar \b'', \bar \b},\xs |\theta^+|\Theta_{\bar \b, \bar \b})\\
=&\left[ h_{\a\to \a'\to \a''}^{\b}(-)|\theta^+|\lambda_{\bar{\b}'', \bar{\b}}+f_{\a'',\a,\b}(\lambda_{\a'',\a},-)|\theta^+|\lambda_{\bar \b'', \bar \b},\d\right](\xs).
\end{split}
\label{eq:hypercube-relations-diagonal-1-handle}
\end{equation}

We note that

\begin{equation}
\begin{split}
&\left[ h_{\a\to \a'\to \a''}^{\b}(-)|\theta^+|\lambda_{\bar{\b}'', \bar{\b}},\d\right](\xs)
\\
=&f_{\a'',\a,\b}(f_{\a'',\a',\a}(\Theta_{\a'',\a'},\Theta_{\a',\a}),\xs)|\theta^+|\lambda_{\bar{\b}'',\bar \b}\\
&+\left(f_{\a'\to \a''}^{\b}\circ f_{\a\to \a'}^{\b}\right)(\xs)|\theta^+|\lambda_{\bar \b'', \bar \b}\\
&+h_{\a\to \a'\to \a''}^{\b}(\xs)|\theta^+| \Theta_{\bar \b'', \bar \b}\\
&+h_{\a\to \a'\to \a''}^{\b}(\xs)|\theta^+| f_{\bar \b'', \bar \b', \bar \b}(\Theta_{\bar \b'', \bar \b'}, \Theta_{\bar \b', \bar \b}).
\end{split}
\label{eq:1-handle-hypercube-simpl-1}
\end{equation}

\begin{equation}
\begin{split}
&\left[f_{\a'',\a,\b}(\lambda_{\a'',\a},-)|\theta^+|\lambda_{\bar \b'', \bar \b},\d\right](\xs)\\
=&f_{\a'', \a, \b}(\Theta_{\a'',\a}+f_{\a'',\a',\a}(\Theta_{\a'',\a'}, \Theta_{\a',\a}),\xs)|\theta^+|\lambda_{\bar \b'', \bar \b}
\\
f_{\a'',\a,\b}(\lambda_{\a'',\a},\xs)|\theta^+|(\Theta_{\bar \b'', \bar \b}+f_{\bar \b'', \bar \b', \bar \b}(\Theta_{\bar \b'', \bar \b'}, \Theta_{\bar \b', \bar \b})).
\end{split}
\label{eq:1-handle-hypercube-simpl-2}
\end{equation}

Next, we note that the stabilization result from Proposition <ref>
implies that

\begin{equation}
\begin{split}f_{\a'' c \bar \b'', \a c \bar \b, \b c \bar \b}(\lambda_{\a'',\a}|\theta^+|f_{\bar \b'', \bar \b', \bar \b}(\Theta_{\bar{\b}'', \bar{\b}'}, \Theta_{\bar \b', \bar \b}), \xs |\theta^+|\Theta_{\bar \b, \bar \b})
&=f_{\a'', \a, \b}(\lambda_{\a'', \a}, \xs)|\theta^+|f_{\bar \b'', \bar \b', \bar \b}(\Theta_{\bar{\b}'', \bar{\b}'}, \Theta_{\bar \b', \bar \b}),
\\
f_{\a'' c \bar \b'', \a c \bar \b, \b c \bar \b}(\Theta_{\a'', \a}|\theta^+|\lambda_{\bar \b'', \bar \b},\xs |\theta^+|\Theta_{\bar \b, \bar \b})
&=f_{\a'', \a, \b}(\Theta_{\a'',\a}, \xs)|\theta^+|\lambda_{\bar \b'', \bar \b}.
\end{split}
\label{eq:1-handle-hypercube-simpl-3}
\end{equation}

Using (<ref>),  (<ref>) and (<ref>) to simplify (<ref>), we see that the length 3 hypercube relation becomes equivalent to

\begin{equation}
\begin{split}
&h_{\a'' c \bar \b'', \a' c \bar \b', \a c\bar \b, \b c \bar \b}(\Theta_{\a'',\a'}|\theta^+|\Theta_{\bar \b'' \bar \b'}, \Theta_{\a',\a}|\theta^+|\Theta_{\bar \b',\bar \b},\xs|\theta^+|\Theta_{\bar \b, \bar \b})\\
=&h_{\a\to \a'\to \a''}^{\b}(\xs)|\theta^+| f_{\bar \b'', \bar \b', \bar \b}(\Theta_{\bar \b'', \bar \b'}, \Theta_{\bar \b', \bar \b}).
\end{split}
\label{eq:hypercube-reformulated-diagonal-1handle}
\end{equation}

We now prove  (<ref>).   Suppose that $\psi$ and $\psi_0$ are classes of rectangles on $(\Sigma,\as'',\as',\as,\bs)$ and $(\bar{\Sigma}, \bar{\bs}'', \bar{\bs}',\bar{\bs}, \bar{\bs})$ such that $\mu(\psi)+\mu(\psi_0)=-1$. Let $C(\psi,\psi_0)$ denote the set of all homology classes which coincide with $\psi$ and $\psi_0$ outside the 1-handle region, and have $\theta^+$ as their input and output in the 1-handle region. (Such rectangles necessarily have index $-1$).  Applying Proposition <ref> to the 1-handle region allows us to remove the 1-handle region connecting $\Sigma$ and $\bar{\Sigma}$. We obtain

\begin{equation}
\sum_{\phi\in C(\psi,\psi_0)} \# \cM(\phi)\equiv \#( \cM(\psi)\times_{\ev_{\Box}} \cM(\psi_0)) \pmod 2 \label{eq:rectangles-and-1-handles}.
\end{equation}

Here, we are also using the nodal almost complex structures around the 1-handle region on the left-hand side. Furthermore any class of rectangles which is not in some $C(\psi,\psi_0)$ for a pair $(\psi,\psi_0)$ satisfying $\mu(\psi)+\mu(\psi_0)=-1$ has no holomorphic representatives. Also, $\times_{\ev_{\Box}}$ denotes the fibered product over the evaluation map to the parameter space $\R\iso \cM(\Box)$.

We write $h^0$ for the map from $\CF^-(\as,\bs)$ to $\CF^-(\as''c\bar{\bs}'',\bs c \bar{\bs})$ which is the composition

\[
h^0=h_{\a c \bar{\b} \to \a' c \bar{\b}'\to \a'' c \bar{\b}''}^{\b c \bar{\b}}\circ F_1^{c \bar{\b}, c \bar{\b} }.
\]

By Equation (<ref>), we may identify $h^0$ as the map which counts holomorphic rectangles on the disjoint union of two diagrams, viewed as a fibered product over $\ev_{\Box}$. We may consider a map $h^t$ for any generic $t\in \R$, which counts curve pairs $(u,u_0)$ on the disjoint union consisting of holomorphic quadrilaterals such that $\ev_{\Box}(u)=\ev_{\Box}(u_0)+t$.

We claim that $h^t=h^{t'}$ for generic $t$ and $t'$ in $\R$. Firstly, we note that since $\bar{\bs}$ appears twice (once as a small Hamiltonian isotopy), the nearest point argument of <cit.>*Proposition 11.5 (cf. <cit.>*Section 3.4)  implies that for suitably small choices of translations, the only rectangles on $\bar{\Sigma}$ with representatives have Maslov index at least 0. In particular all holomorphic rectangles counted by $h^t$ for generic $t$ have $\mu(\psi)=-1$ and $\mu(\psi_0)=0$.

We consider the ends of the moduli spaces counted by $h^t$, ranging over $t\in [0,\infty)$. No degenerations are possible on the $\Sigma$-side, since all rectangles there have index $-1$. On the $\bar{\Sigma}$-side, rectangles have index $0$, so the possible degenerations are a holomorphic disk splitting off at some finite $t$, as well as the ends which appear as $t\to \infty$. The ends appearing at $t\to \infty$ correspond exactly to the map

\[
h_{\a\to \a'\to \a''}^{\b}(\xs)|\theta^+|f_{\bar{\b}'', \bar{\b}',\bar{\b}}(\Theta_{\bar{\b}'', \bar{\b}'}, f_{\bar{\b}', \bar{\b}, \bar{\b}}(\Theta_{\bar{\b}', \bar{\b}}, \Theta_{\bar{\b},\bar{\b}})).
\]

The ends at finite $t$ are slightly more subtle. First, one end which can appear is an end where a disk breaks off into the $\bar{\bs}$-$\bar{\bs}$ end. These cancel modulo two since $\Theta_{\bar{\b}, \bar{\b}}$ is a cycle. Holomorphic disks breaking off into the other ends are prohibited, since they would leave an index $-1$ holomorphic rectangle, with $\Theta_{\bar{\b},\bar{\b}}$ as input. (The most important end where this is prohibited is the outgoing $\bar{\bs}''$-$\bar{\bs}$ end, where a more general diagram might have degenerations.) In particular, we obtain (<ref>), completing the proof.

The hypercube in the bottom right of Figure <ref> is constructed by simply interchanging the roles of the alpha and beta curves in the above procedure.

We now construct the maps in the top right hypercube of Figure <ref>. As a first step, we pick chains to give hypercubes of attaching curves, as follows:

* The top face is obtained by pairing the two hypercubes

\[
\begin{tikzcd}
\as'\ar[r, "\Theta_{\a'',\a'}"]
\as''
\end{tikzcd}
\quad \text{and} \quad
\begin{tikzcd}
\bs\ar[r, "\Theta_{\b,\b'}"]
\bs'
\end{tikzcd}
\]

* The bottom face is obtained by pairing the two hypercubes

\[
\begin{tikzcd}[column sep =2.5cm]
\as'c \bar{\bs}'\ar[r, "\Theta_{\a'',\a'}|\theta^+|\Theta_{\bar \b'', \bar \b'}"]
\as''c \bar{\bs}''
\end{tikzcd}
\quad \text{and} \quad
\begin{tikzcd}[column sep =2.5cm]
\bs c \bar \bs\ar[r, "\Theta_{\b,\b'}|c|\Theta_{\bar \b, \bar \b'}"]
\bs' c \bar \bs'
\end{tikzcd}
\]

* The length 2 map along the left face is

\[
\xs\mapsto f_{\a'\to \a''}^{\b}(\xs)| \theta^+| \lambda_{\bar{\b}'',\bar{\b}}.
\]

* The length 2 map along the front face is

\[
\xs\mapsto f_{\a''}^{ \b\to \b'}(\xs)| \theta^+| \lambda_{\bar \b'', \bar \b'}.
\]

* The length 3 map is the sum of the following two maps

\[
\begin{split}
\xs&\mapsto \left(f_{\a''}^{\b\to \b'}\circ f_{\a'\to \a''}^{\b}\right)(\xs)|\theta^+|\omega_{\bar \b'', \bar \b'}\\
\xs&\mapsto h^{t<t_0}(\xs)
\end{split}
\]

The map $h^{t<t_0}$ is a new map, as follows. It counts pairs $(u,u_0)$, where $u$ and $u_0$ are holomorphic rectangles on $(\Sigma,\as'',\as',\bs,\bs')$ and $(\bar{\Sigma}, \bar \bs'',\bar \bs',\bar\bs,\bar \bs')$, respectively, both of index $-1$, such that

\[
\ev_{\Box}(u)<\ev_{\Box}(u_0).
\]

If $\ys$ and $\zs$ are the outputs of $u$ and $u_0$, respectively, then the $h^{t<t_0}(\xs)$ has a summand of $\ys\times \theta^+\times \zs$ (weighted by the appropriate $U$-power).

With the maps described above, the top right subcube of Figure <ref> is a hypercube of chain complexes.

The proof is similar to the proof of Lemma <ref>. The length 1 and 2 hypercube relations are similar to the previous case, so we focus on the length 3 relation. If $\xs\in \bT_{\a'}\cap \bT_{\b}$, the desired relation is

\begin{equation}
\begin{split}
&h_{\a'\to \a''}^{\b\to \b'}(\xs)|\theta^+|\Theta_{\bar \b'', \bar\b'}\\
&+f_{\a'' c \bar \b'', \b c \bar \b,\b' c \bar \b'}\left(f_{\a'\to \a''}^{\b}(\xs)|\theta^+|\lambda_{\bar \b'', \bar \b}, \Theta_{\b, \b'}|\theta^+|\Theta_{\bar \b, \bar \b'}\right)\\
&+\left(f_{\a''}^{\b\to \b'}\circ f_{\a'\to \a''}^{\b}\right) (\xs)|\theta^+|\lambda_{\bar \b'', \bar \b'}\\
&+h_{\a' c \bar \b'\to \a'' c \bar \b''}^{\b c \bar \b\to \b' c \bar \b'}(\xs|\theta^+|\Theta_{\bar \b', \bar \b})\\
=&\left[\left(f_{\a''}^{\b\to \b'}\circ f_{\a'\to \a''}^{\b}\right)(-)|\theta^+|\omega_{\bar \b'', \bar \b'}+h^{t<t_0}, \d \right](\xs)
\end{split}
\label{eq:length-3-relation-well-defined-ab}
\end{equation}

By excising the special curves in the connected sum region using the results about holomorphic triangles and 1-handles from <cit.>*Theorem 8.8 (compare Proposition <ref>, above), we obtain

\begin{equation}
\begin{split}
&f_{\a'' c \bar \b'', \b c \bar \b,\b' c \bar \b'}\left(f_{\a'\to \a''}^{\b}(\xs)|\theta^+|\lambda_{\bar \b'', \bar \b}, \Theta_{\b, \b'}|\theta^+|\Theta_{\bar \b, \bar \b'}\right)\\
=&\left(f^{\b\to \b'}_{\a''}\circ f_{\a' \to \a''}^{\b}\right)(\xs)|\theta^+|f_{\bar \b''}^{\bar \b \to \bar \b'}(\lambda_{\bar \b'', \bar \b}).
\end{split}
\label{eq:hyperbox-holomorphic-triangles-ab-well-def}
\end{equation}

We now analyze the quadrilateral map $h_{\a' c \bar \b'\to \a'' c \bar \b''}^{\b c \bar \b\to \b' c \bar \b'}(\xs|\theta^+|\Theta_{\bar \b', \bar \b})$. As with the previous case, we can destabilize the 1-handle region and identify the expression with the count of index $-1$ holomorphic quadrilaterals on $(\Sigma\sqcup \bar \Sigma)\times \Box$. We can view this moduli space as the fibered product over the evaluation map to $\cM(\Box)\iso \R$. If $(\psi,\psi_0)$ is a homology class on $(\Sigma \sqcup \bar \Sigma)\times \Box$ which admits a holomorphic representative, then there are generically two possibilities:

*  $\mu(\psi)=0$ and $\mu(\psi_0)=-1$;

*  $\mu(\psi)=-1$ and $\mu(\psi_0)=0$.

As before we will write $h^0$ for the map $h_{\a' c \bar \b'\to \a'' c \bar \b''}^{\b c \bar \b\to \b' c \bar \b'}(\xs|\theta^+|\Theta_{\bar \b', \bar \b})$. For $t\in [0,\infty)$, we write $h^t$ for a deformation, which counts curves such that

\[
\ev_{\Box}(u_0)-\ev_{\Box}(u)=t.
\]

We count the ends of the moduli spaces defining the maps $h^t$ (ranging over $t\in [0,\infty)$). We obtain

\begin{equation}
\begin{split}
[\d, h^{t<t_0}](\xs)=&h_{\a' c \bar \b'\to \a'' c \bar \b''}^{\b c \bar \b\to \b' c \bar\b'}(\xs|\theta^+|\Theta_{\bar \b', \bar \b})\\
&+(f^{\b\to \b'}_{\a''}\circ f_{\a'\to \a''}^{\b})(\xs)|\theta^+|h_{\bar \b'\to \bar \b''}^{\bar \b\to \bar \b'}(\Theta_{\bar \b', \bar \b})\\
&+h_{\a'\to \a''}^{\b\to \b'}(\xs)|\theta^+|\Theta_{ \bar \b'', \bar \b'}.
\end{split}
\label{eq:deform-diagonals-quadrilaterals}
\end{equation}

Note that  instead of $\Theta_{\bar \b'', \bar \b'}$ in the final factor, what more  naturally appears is

\[
f_{\bar \b'', \bar \b', \bar \b'}(\Theta_{\bar \b'', \bar \b'}, f_{\bar \b', \bar \b, \bar \b'}(\Theta_{\bar \b', \bar \b}, \Theta_{\bar \b, \bar \b'})).
\]

However this coincides with $\Theta_{\bar \b'', \bar \b'}$, by a nearest point computation, cf. <cit.>*Proposition 11.1.

\begin{equation}
\begin{split}
&\left[\left(f_{\a''}^{\b\to \b'}\circ f_{\a'\to \a''}^{\b}(-)\right)|\theta^+|\omega_{\bar \b'', \bar \b'}, \d \right](\xs)\\
=&\left(f_{\a''}^{\b\to \b'}\circ f_{\a'\to \a''}^{\b}(\xs)\right)|\theta^+|\left(f_{\bar \b''}^{\bar \b \to \bar \b'}(\lambda_{\bar \b'', \bar \b})+\lambda_{\bar \b'', \bar \b'}+h_{\bar \b'\to \bar \b''}^{\bar \b \to \bar \b'}(\Theta_{\bar \b', \bar \b})\right)
\end{split}
\label{eq:def-omega-b''b'}
\end{equation}

By combining  (<ref>),  (<ref>) and  (<ref>), we quickly obtain  (<ref>), completing the proof.

§ 1- AND 3-HANDLES

In this section, we define our cobordism maps for 1-handle and 3-handles. Since for the maps in this section there will be an obvious choice of $\Spin^c$ structure and framed path, we omit these from the notation throughout.

§.§ The construction

We now construct the map for a 4-dimensional 1-handle. Suppose that $\bS_0=\{p,p'\}$ is a (framed) 0-sphere in $Y$. We pick a Heegaard surface $(\Sigma,\as,\bs)$ so that $\bS_0\subset \Sigma\setminus (\as\cup \bs)$. We form a Heegaard surface for the surgered manifold $Y(\bS_0)$ by cutting out small disks centered at $p$ and $p'$, and gluing in an annular diagram $(A,\g_0,\g_0)$, where $\g_0$ and $\g_0$ are meridians of the annulus which intersect in two points. (Here and onwards, we abuse notation and view one copy of $\g_0$ as a Hamiltonian translate of the other). Write $(\Sigma', \as \g_0, \bs \g_0,w)$ for the diagram obtained by this procedure.

We now define the 4-dimensional 1-handle map. To do this, we first pick paths $\lambda$ and $\lambda'$ from $p$ and $p'$ to $w$. We pick doubling arcs $\ds$ for $\Sigma$ which are disjoint from $\lambda$ and $\lambda'$. Furthermore, we choose two more doubling arcs $\ds_0$ on $\Sigma'$, which are contained in the union of a neighborhood of $\lambda$ and $\lambda'$, as well as the annular 1-handle region. We assume $\ds\cup \ds_0$ is a valid collection of doubling arcs for $\Sigma'$.  Write $\Ds$ and $\Ds_0$  for the curves obtained by doubling $\ds$ and $\ds_0$, respectively.

Additionally, the involutive 1-handle map requires a choice of closed curve $\tau\subset \Sigma'\# \bar \Sigma'$, such that the path $\tau$ is obtained by doubling a properly embedded arc on $\Sigma'\setminus N(w)$ which intersects $\g_0$ transversely in a single point.

We define the 1-handle map to be the diagram obtained by compressing the following hyperbox:

\begin{equation}
\begin{tikzcd}[
row sep=1cm,
column sep=2.4cm,
fill opacity=.7,
text opacity=1
\CF(\as,\bs)
\ar[d, "F_{1}^{\bar{\b},\bar{\b}}"]
\ar[rrr, "F_1^{\g_0,\g_0}"]
\CF(\as\g_0,\bs\g_0)
\ar[d,"F_{1}^{\bar{\b}\bar \g_0,\bar{\b} \bar \g_0}"]
\\
\CF(\as \bar{\bs}, \bs \bar{\bs})
\ar[rr, "F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[d,"f^{\b \bar{\b}\to \Dt}_{\a \bar{\b}}"]
\CF(\as \bar \bs \g_0 \bar \g_0, \bs  \bar \bs \g_0  \bar \g_0)
\ar[r,equal]
\ar[d, "f^{\b \bar \b\g_0 \bar \g_0\to \Dt \g_0 \bar \g_0}_{\a \bar \b \g_0 \bar \g_0}"]
\ar[dr,dashed]
& \CF(\as\bar \bs \g_0 \bar \g_0, \bs \bar \bs \g_0 \bar \g_0)
\ar[d,"f^{\b \bar \b \g_0 \bar \g_0\to \Dt \Dt_0}_{\a \bar \b \g_0 \bar \g_0}"]
\\
\CF(\as \bar \bs,\Ds)
\ar[rr, "F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[d, "f^{\Dt\to \a  \bar{\a}}_{\a  \bar{\b}}"]
\CF(\as \bar \bs  \g_0 \bar \g_0, \Ds \g_0 \bar \g_0)
\ar[r, "f_{\a \bar \b \g_0 \bar \g_0}^{\Dt \g_0 \bar \g_0 \to \Dt \Dt_0}"]
\ar[d, "f_{\a \bar \b \g_0 \bar \g_0}^{\Dt \g_0 \bar \g_0\to \a \bar \a \g_0 \bar \g_0}"]
\ar[dr,dashed]
& \CF(\as\bar{\bs} \g_0 \bar \g_0,\Ds \Ds_0)
\ar[d, "f_{\a \bar \b \g_0 \bar \g_0}^{\Dt \Dt_0\to \a \bar \a \g_0 \bar \g_0}"]
\\
\CF(\as  \bar{\bs}, \as  \bar{\as})
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
\ar[d, "F_{3}^{\a,\a}"]
\CF(\as \bar \bs \g_0 \bar \g_0, \as \bar \as \g_0 \bar \g_0)
\ar[r, "B_\tau"]
\CF(\as \bar \bs \g_0 \bar \g_0, \as \bar \as \g_0 \bar \g_0)
\ar[d, "F_3^{\a\g_0,\a\g_0}"]
\\
\CF(\bar{\bs}, \bar{\as})
\ar[rrr, "F_1^{\bar \g_0, \bar \g_0}"]
\CF(\bar{\bs} \bar{\g}_0,\bar{\as} \bar{\g}_0)
\end{tikzcd}
\label{eq:1-handles}
\end{equation}

It will be helpful to observe that the bottom-most level of (<ref>) has the following slightly expanded form

\begin{equation}
\begin{tikzcd}[
row sep=1cm,
column sep=1.5cm,
fill opacity=.7,
text opacity=1
\CF(\as  \bar{\bs}, \as  \bar{\as})
\ar[r, "F_1^{\bar \g_0, \bar \g_0}"]
\ar[d, equal]
\CF(\as\bar \bs\bar \g_0, \as \bar \as\bar \g_0)
\ar[r,"F_1^{\g_0,\g_0}"]
\ar[d,equal]
\CF(\as \bar \bs \g_0 \bar \g_0, \as \bar \as \g_0 \bar \g_0)
\ar[r, "B_\tau"]
\CF(\as \bar \bs \g_0 \bar \g_0, \as \bar \as \g_0 \bar \g_0)
\ar[d, "F_3^{\g_0,\g_0}"]
\\
\CF(\as\bar{\bs}, \as\bar{\as})
\ar[r, "F_1^{\bar \g_0, \bar \g_0}"]
\ar[d, "F_3^{\a,\a}"]
\CF(\as\bar \bs \bar \g_0,\as \bar \as \bar \g_0)
\ar[rr,equal]
\CF(\as\bar{\bs} \bar{\g}_0,\as\bar{\as} \bar{\g}_0)
\ar[d,"F_3^{\a,\a}"]
\\
\CF(\bar{\bs}, \bar{\as})
\ar[rrr, "F_1^{\bar \g_0, \bar \g_0}"]
\CF(\bar{\bs} \bar{\g}_0,\bar{\as} \bar{\g}_0)
\end{tikzcd}
\label{eq:1-handles-expanded-bottom-portion}
\end{equation}

We now describe the construction of (<ref>).  The top-most face which has a diagonal map is obtained by pairing a hypercube of beta attaching curves with the hypercube of alpha attaching curves consisting of  $\as\bar \bs \g_0\bar \g_0$.

We now consider the lowest face of (<ref>) which has a diagonal map. This map is obtained by pairing the 0-dimensional alpha hypercube $\as \bar \bs \g_0\bar \g_0$ with the diagram below. The map $B_\tau$ is the $H_1$-action for $\tau$, defined by counting changes across the beta curves, $\as\bar \as \g_0\bar \g_0$.

\begin{equation}
\begin{tikzcd}[labels=description, row sep =2cm, column sep =2.5cm]
\Ds \g_0\bar \g_0
\ar[r,"\Theta_{\Dt \g_0\bar \g_0, \Dt \Dt_0}"]
\ar[d,"\Theta_{\Dt \g_0\bar \g_0, \a \bar \a \g_0\bar \g_0}"]
\ar[dr, dashed, "\eta"]
\Ds \Ds_0
\ar[d, "\Theta_{\Dt \Dt_0, \a \bar \a \g_0 \bar \g_0}"]
\\
\as \bar \as \g_0 \bar \g_0
\ar[r ,"\tau"]
\as \bar \as \g_0 \bar \g_0
\end{tikzcd}
\label{eq:1-handle-hypercube-involving-H1}
\end{equation}

In equation (<ref>), the hypercube relations are to be interpreted using the formalism of Section <ref>. In particular, the hypercube relations amount to the equation

\begin{equation}
\d \eta=B_\tau(\Theta_{\Dt\g_0\bar \g_0, \a \bar \a \g_0\bar \g_0})+f_{\Dt \g_0\bar \g_0, \Dt \Dt_0, \a \bar \a\g_0\bar \g_0}(\Theta_{\Dt \g_0\bar \g_0, \Dt\Dt_0}, \Theta_{\Dt\Dt_0,\a \bar \a \g_0\bar \g_0}).
\label{eq:1-handle-hypercube-involving-H1-unpacked}
\end{equation}

There are many choices of morphisms in the diagram in equation (<ref>). Any two choices of top-degree generators $\Theta_{\Dt \g_0 \bar \g_0, \Dt \Dt_0}$ are homologous. Similarly any two choices of $\Theta_{\Dt \g_0 \bar \g_0, \a \bar\a \g_0 \bar \g_0}$ and $\Theta_{\Dt \Dt_0, \a \bar \a \g_0 \bar \g_0}$ are homologous. Unlike these morphisms, it is not the case that any two choices of diagonal morphism $\eta=\eta_{\Dt \g_0\bar \g_0, \a \bar \a \g_0 \bar \g_0}$ are homologous. Indeed this chain lies in the top degree of $\CF^-(\#^{g+1} S^1\times S^2)$, and hence there two choices which are not homologous. Similarly, it is not the case that any two choices for the curve $\tau$ are homologous. Nonetheless, we will show in Lemma <ref> that resulting involutive 1-handle map obtained by compressing the diagram in equation (<ref>) is independent from these choices, up to chain homotopy.

We now show that this is satisfiable:

There is a homogeneously graded $\eta\in \CF(\Ds\g_0\bar \g_0, \as \bar \as \g_0\bar \g_0)$ satisfying (<ref>).

Any two choices of $\Ds\Ds_0$ are related by a sequence of handleslides and isotopies. Via an associativity argument, it is sufficient to show the claim for any convenient choice of $\Ds$ and $\Ds_0$. In particular, we may assume that the curves of $\Ds_0$ are disjoint from $\as$ and $\bar \as$. In particular, the relevant triple diagram $(\Sigma' \# \bar \Sigma', \Ds \g_0 \bar \g_0,\Ds \Ds_0,\as \bar \as \g_0 \bar \g_0)$ may be decomposed as a connected sum

\[
(\Sigma\# \bar \Sigma,\Ds,\Ds,\as \bar \as)\#(\bT^2\# \bT^2,\g_0 \bar \g_0, \Ds_0, \g_0 \bar \g_0).
\]

By the stabilization result in Proposition <ref>, it suffices to show the claim for the genus 2 triple $(\bT^2\# \bT^2,\g_0 \bar \g_0, \Ds_0, \g_0 \bar \g_0)$.

There are two ways to verify the claim in this situation. The first strategy is to argue topologically via the following reasoning:

* Topologically the triple $(\bT^2\# \bT^2,\g_0 \bar \g_0, \Ds_0, \g_0 \bar \g_0)$ is for 0-surgery on an unknot in $S^1\times S^2$.

* By picking $\Ds_0$ appropriately, we may assume that $\tau$ runs parallel to a curve $\Dt\in  \Ds_0$. Furthermore, $\Dt$ is the meridian of the knot on which we are performing surgery, and in particular defines the dual of the surgery knot.

* In general, if $U$ is a 0-framed unknot in $Y$, and $\mu$ is a meridian, then $\CF(W(U))=A_\mu\circ F_{1}$, where $F_1$ is the 1-handle map. This is easily verified using a genus 1 diagram and a stabilization result for triangles.

* By combining the ideas above, the proof is complete.

Alternately, one may explicitly perform the computation by counting holomorphic triangles. Write $\theta^{\pm}$ (resp. $\bar{\theta}{}^{\pm}$) for the intersection points of $\g_0$ with its translate (resp. $\bar \g_0$ with its translate). In Figure <ref>, we show the four index 0 triangle classes which can contribute.
Two of the classes are in $\pi_2(\Theta^+,\Theta^+,\theta^+|\bar{\theta}{}^-)$ and two of the classes are in $\pi_2(\Theta^+,\Theta^+,\theta^-|\bar{\theta}{}^+)$.
If $\psi_1$ and $\psi_2$ denote the two classes in $\pi_2(\Theta^+,\Theta^+,\theta^+|\bar{\theta}{}^-)$, we claim

\begin{equation}
\# \cM(\psi_1)+\# \cM(\psi_2)\equiv 1\mod 2, \label{eq:sum-of-classes-is-1}
\end{equation}

and similarly for the two classes in $\pi_2(\Theta^+,\Theta^+,\theta^-|\bar{\theta}{}^+)$.
This claim is verified via a Gromov compactness argument which is illustrated and explained in Figure <ref>.
Hence $f_{\g_0\bar \g_0, \Dt_0, \g_0 \bar \g_0}(\Theta^+,\Theta^+)=\theta^+|\bar{\theta}^-+\theta^-|\bar{\theta}^+$, which is also easily seen to be the action $\tau$ on $\theta^+|\bar \theta^+$.

Verifying equation (<ref>) by counting triangles. Left: The diagram $(\bT^2\# \bT^2, \g_0\bar\g_0, \Dt_0)$. Right: The four classes of index 0 holomorphic triangles which contribute to the triangle map $f_{\g_0\bar \g_0,\Dt_0,\g_0\bar\g_0}(\Theta^+,\Theta^+)$.

A Gromov compactness argument which shows that the total count of holomorphic triangles of index 0 in $\pi_2(\Theta^+,\Theta^+,\theta^+|\bar{\theta}{}^-)$ is 1. In the left-most columns are index 1 classes of triangles. On the other columns of a given row, we show all decompositions of that class into an index 0 triangle class and an index 1 disk class (such that both classes have nonnegative multiplicities). Configurations $A$ and $E$ have the same total count as the $\theta^+|\bar{\theta}{}^-$ coefficient of our map. Configuration $B$ has total count 1. Configurations $C$ and $D$ have the same count. Gromov compactness gives $\# A+\# B+\# C\equiv 0$ and $\# D+\# E\equiv 0$, which quickly gives Equation (<ref>).

Finally, we verify that the bottom-most face of (<ref>) is a hypercube. Note that this face may be expanded into the diagram shown in (<ref>). In light of this, it suffices to show the following relation (note the similarity to Equation  (<ref>)):

As maps from $\CF(\as \bar \bs \bar \g_0, \as \bar \as \bar \g_0)$ to $\CF(\as \bar \bs \bar \g_0, \as \bar \as \bar \g_0)$, there is an equality

\[
F_3^{\g_0,\g_0}\circ B_\tau \circ F_1^{\g_0,\g_0}=\id.
\]

Compare <cit.>*Lemma 7.10. The relation follows from the form of the differential after attaching a 1-handle. Namely, it follows from <cit.>*Proposition 8.5 that the differential satisfies

\[
\d(\xs\times \theta^+_{\g_0,\g_0})= \d(\xs)\otimes \theta^+_{\g_0,\g_0}.
\]

Furthermore, the proof shows that the holomorphic curves which go from $\xs\times \theta^+_{\g_0,\g_0}$ to $\ys\times \theta^-_{\g_0,\g_0}$ for some $\xs$ and $\ys$ all have $\xs=\ys$ and have domain consisting of one of the two bigons contained in the 1-handle region. The 1-chain $\tau$ intersects the boundary of exactly one of these bigons, and hence $B_{\tau}(\xs\times \theta_{\g_0,\g_0}^+)= \xs\times \theta_{\g_0,\g_0}^-+\zs\otimes \theta_{\g_0,\g_0}^+$, for some $\zs\in \CF(\as \bar \bs, \as \bar \as)$. The stated claim follows immediately.

§.§ Well-definedness of the 1-handle maps

In this section, we prove that the stabilization maps are compatible with the maps for changing the alpha and beta curves via handleslides or isotopies, or changing the choice of doubling curves $\Ds$ and $\Ds_0$. We also prove that the involutive 1-handle maps are independent of the choice of the curve $\tau\subset \Sigma'\# \bar \Sigma'$, as well as the choice of chains used to construct the hypercubes in (<ref>).

As a first step, we verify independence from the curve $\tau$:

Suppose that $\tau$ and $\tau'$ are two closed curves on $\Sigma'\# \bar \Sigma'$ obtained by doubling  properly embedded arcs on $\Sigma'\setminus N(w)$ which intersect $\g_0$ transversely in a single point. Then there are choices of the additional auxiliary data in the construction so that the models of the 1-handle maps in Equation (<ref>) are chain homotopic.

In the compression of Equation (<ref>) the choice of $\tau$ only affects the diagonal map. There are two ways in which it appears. The first is via the choice of $\eta$, and the second is a summand which has a factor of

\begin{equation}
F_3^{\a \g_0,\a \g_0}\circ f_{\a \bar \b \g_0\bar \g_0}^{\Dt \g_0\bar \g_0\to \a \bar \a \g_0\bar \g_0\xrightarrow{\tau} \a \bar \a \g_0\bar \g_0}\circ F_{1}^{\g_0\bar \g_0, \g_0\bar \g_0}.\label{eq:1-handle-map-tau-contribution}
\end{equation}

We consider this latter summand first. The map in Equation (<ref>) counts holomorphic triangles weighted by $\# \tau \cap \d_{\a \bar \a \g_0\bar \g_0}(\psi)$. If $\psi\in \pi_2(\xs,\ys,\zs)$ is a class of triangles counted in the above composition, then both the $\xs$ and $\ys$ (the incoming intersection points) have $\theta_{\g_0,\g_0}^+$ as factors. By the stabilization result of Proposition <ref>, if such a class has a representative, then $\zs$ must have a factor of $\theta_{\g_0,\g_0}^+$ as well. However, such a term would evaluate trivially under $F_3^{\a\g_0,\a \g_0}$. Hence, Equation (<ref>) vanishes.

We now consider the first summand, which involves the choice of $\eta$. Let $\tau$ and $\tau'$ be two choices of curves, as above, and let $\eta_\tau$ be some choice of diagonal chain for the hypercube from Equation (<ref>), when $\tau$ is used. It is straightforward to see that $[\tau]-[\tau']\in H_1(\Sigma'\# \bar \Sigma')$ lies in the span of the $\Ds$ curves.  Let $C$ be a 2-chain on $\Sigma$ such that $\d \Sigma$ is $\tau-\tau'$ plus a linear combination of small parallel pushoffs of the $\Ds$ curves.

We define the 2-chain

\[
\eta_{\tau'}:=\eta_\tau+H_C(\Theta_{\Dt \g_0\bar \g_0, \a \bar \a \g_0\bar \g_0}).
\]

In the above, $H_C$ denotes the $\bF[U]$-linear map whose value on an intersection point $\xs\in \bT_{\Dt\g_0\bar \g_0}\cap \bT_{\a \bar \a \g_0\bar \g_0}$ is the sum of the multiplicities of $C$ at each factor of $\xs$. It is straightforward to see that $\eta_{\tau'}$ is a valid choice of diagonal for the hypercube in Equation (<ref>) which uses $\tau'$.
Since each summand of $H_C(\Theta_{\Dt \g_0\bar \g_0, \a \bar \a \g_0\bar \g_0})$ has $\theta_{\g_0,\g_0}^+$ as a factor, the same argument as above shows that

\[
F_3^{\g_0,\g_0}\circ f_{\a \bar \b\g_0\bar \g_0, \Dt \g_0\bar \g_0, \a \bar \a \g_0\bar \g_0}(F_1^{\g_0,\g_0}(\xs), H_C(\Theta_{\Dt \g_0\bar \g_0, \a \bar \a \g_0\bar \g_0}))=0,
\]

so the diagonal maps computed using $\eta_\tau$ and $\eta_{\tau'}$ coincide.

Suppose that $\cH=(\Sigma,\as,\bs)$ and $\cH'=(\Sigma,\as',\bs')$ are two Heegaard diagrams for $Y$ with the same underlying Heegaard surface, such that $\as'$ is obtained from $\as$ by an elementary handleslide or isotopy. We assume $\bs'$ is obtained similarly from $\bs$. Let $\frD$ and $\frD'$ denote doubling enhancements of $\cH$ and $\cH'$ with choices of almost complex structures and doubling curves $\Ds$ and $\Ds'$.  Let $\bS_0$ denote a 0-sphere which is embedded in $\Sigma\setminus (\as\cup \bs \cup \as'\cup \bs')$ and let $\Sigma(\bS_0)$ denote the result of attaching a 1-handle along $\bS_0$ to $\Sigma$. Let $\Ds_0$ and $\Ds_0'$ denote extra choices of doubling arcs, as in the construction of the 1-handle map, and let $\frD(\bS_0)$ and $\frD'(\bS_0)$ denote the doubling enhancements of $(\Sigma(\bS_0),\as\g_0,\bs\g_0)$ and $(\Sigma(\bS_0),\as'\g_0,\bs'\g_0)$ by using the new doubling curves $\Ds_0$ and $\Ds_0'$, respectively. Let $\tau$ be a single choice of symmetric 1-cycle on $\Sigma(\bS_0)\# \bar \Sigma(\bS_0)$ which intersects $\g_0$ transversely at a single point. Finally, let $\CFI(W(\bS_0))$ and $\CFI(W(\bS_0))'$ denote the involutive 1-handle maps, computed with these two choices of data. Then

\begin{equation}
\Psi_{\frD(\bS_0)\to \frD'(\bS_0)}\circ \CFI(W(\bS_0))\simeq \CFI(W(\bS_0))'\circ \Psi_{\frD\to \frD'}.
\label{eq:transition-map-commutes-1-handles}
\end{equation}

The proof is to enlarge the diagram in (<ref>) into a 3-dimensional hyperbox by adding an extra direction coming out of the page. One face will correspond to the map $\CFI(W(\bS_0))$ and an opposite face will correspond to the map $\CFI(W(\bS_0))'$. It will be straightforward to unpack the hypercube relations to obtain (<ref>). We presently describe the construction of the hyperbox.

We extend the top-most hypercube of (<ref>) to the hyperbox shown in equation (<ref>). The unlabeled arrows are holomorphic triangle maps. There are no morphisms of length 2 or 3. The hypercube relations follow from the stabilization formula of Proposition <ref>.

\begin{equation}
\begin{tikzcd}[
column sep={1.6cm,between origins},
row sep=1cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;
\CF(\as ,\bs)
\ar[dr]
\ar[ddd, "F_1^{\bar \b,\bar \b}"]
\ar[rr,"F_1^{\g_0,\g_0}"]
&[.7 cm]
&\CF(\as\g_0,\bs \g_0)
\ar[dr]
\ar[ddd, "F_1^{\bar \b \bar\g_0, \bar \b \bar\g_0}"]
&[.7 cm]
\\
&[.7 cm]
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr,"F_1^{\g_0,\g_0}"]
&[.7 cm]
\CF (\as' \g_0, \bs\g_0)
\ar[ddd,swap, pos=.35, "F_1^{\bar \b'\bar \g_0, \bar \b\bar \g_0}"]
\\
\\
\CF(\as \bar \bs, \bs \bar \bs)
\ar[dr]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
&[.7 cm]\,&
\CF(\as \bar \bs \g_0 \bar \g_0, \bs \bar \bs \g_0 \bar \g_0)
\ar[dr]
\\
& [.7 cm]
\CF(\as' \bar \bs', \bs \bar \bs)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu, "F_1^{\bar \b', \bar \b}", pos=.35]
\ar[rr, "F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
&[.7 cm]
\CF(\as' \bar \bs'\g_0\bar \g_0, \bs \bar \bs\g_0\bar \g_0)
\end{tikzcd}
\hspace{-1.7cm}
\begin{tikzcd}[
column sep={1.6cm,between origins},
row sep=1cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;
\CF(\as',\bs)
\ar[dr]
\ar[ddd, "F_1^{\bar \b', \bar \b}"]
\ar[rr, "F_1^{\g_0,\g_0}"]
&[.7 cm]
\CF(\as' \g_0, \bs \g_0)
\ar[rd]
\ar[ddd, "F_1^{\bar \b'\bar \g_0,\bar \b \bar \g_0}"]
&[.7 cm]
\\
&[.7 cm]
\CF(\as',\bs')
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr,"F_1^{\g_0,\g_0}"]
&[.7 cm]
\CF (\as'\g_0,\bs'\g_0)
\ar[ddd, "F_1^{\bar \b' \bar \g_0, \bar \b'\bar \g_0} " ]
\\
\\
\CF(\as'\bar \bs', \bs \bar \bs)
\ar[dr]
\ar[rr,"F_1^{\g_0\bar \g_0,\g_0 \bar\g_0}"]
&[.7 cm]
\CF(\as' \bar \bs' \g_0 \bar \g_0, \bs \bar \bs \g_0 \bar \g_0)
\ar[dr]
\\
& [.7 cm]
\CF(\as'\bar \bs', \bs' \bar \bs')
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu, "F_1^{\bar \b',\bar \b'}", pos=.4]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0 \bar\g_0}"]
&[.7 cm]
\CF(\as' \bar \bs' \g_0 \bar \g_0, \bs' \bar \bs'\g_0 \bar \g_0)
\end{tikzcd}
\label{eq:hyperbox-1-handles-top}
\end{equation}

The hypercubes obtained by extending the two small left-most cubes of (<ref>) are similarly defined to be hyperboxes of stabilizations.

We now consider the right-most cubes of (<ref>) (i.e. the two cubes which have length 2-maps). The top-most cube is easily extended by building hypercubes of attaching curves where each length 1 chain is a cycle generating the top degree of $\HF^-$. We leave the details to the reader.

We now consider the bottom-most cube of (<ref>) which has a length 2 map, and we describe its extension to a 3-dimensional hyperbox. (Recall that this hypercube involved more choices than usual, such as the choice of the chain $\eta$ and the curve $\tau$; see Remark <ref>.) The construction is obtained by pairing hypercubes of attaching curves. The main technical challenge in extending the 2-dimensional cube in (<ref>) is building a 3-dimensional hypercube of attaching curves with the following form:

\begin{equation}
\begin{tikzcd}[
column sep={1.3cm,between origins},
row sep=.8cm,
fill opacity=.7,
text opacity=1,
\Ds \g_0 \bar \g_0
\ar[dr]
\ar[ddd]
\ar[rr]
\ar[dddrr,dashed, "\eta"]
\ar[ddddrrr,dotted]
&[.7 cm]
& \Ds \Ds_0
\ar[rd]
\ar[ddd]
\ar[ddddr,dashed]
&[.7 cm]
\\
&[.7 cm]
\Ds' \g_0 \bar \g_0
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr]
&[.7 cm]
\Ds' \Ds_0'
\ar[ddd]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed]
\\
\\
\as \bar \as \g_0 \bar \g_0
\ar[dr]
\ar[rr,"\tau"]
\ar[drrr,dashed, "h"]
&[.7 cm]\,&
\as \bar \as \g_0 \bar \g_0
\ar[dr]
\\
& [.7 cm]
\as' \bar \as' \g_0 \bar \g_0
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu]
\ar[rr, "\tau"]
\ar[from=uuuul,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuul,dashed]
&[.7 cm]
\as'\bar \as' \g_0 \bar \g_0
\ar[from=uuull,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuull,dashed, "\eta'"]
\end{tikzcd}
\label{eq:hypercube-attaching-curves-1-handle-w-tau}
\end{equation}

In equation (<ref>),  the chains $\eta$ and $\eta'$ are the ones used to construct $\CFI(W(\bS_0))$ and $\CFI(W(\bS_0))'$, as constructed in Lemma <ref>. In particular, these morphisms are fixed, and similarly the top length 2 morphism has also already been chosen in the previous step, so we assume that chain is fixed. Note that the two morphisms from $\as \bar \as \g_0\bar\g_0$ to $\as'\bar \as' \g_0\bar \g_0$ are forced to coincide, since $\as'$ is obtained from $\as$ by an elementary handleslide, so there is a single top degree generator of $\CF(\as'\bar \as'\g_0\bar \g_0, \as \bar \as\g_0\bar \g_0)$, which is $\Theta_{\a'\bar \a', \a \bar \a}\otimes \theta_{\g_0\bar \g_0, \g_0\bar \g_0}^+$.

Assuming for the moment the existence of the hypercube in (<ref>), one extends the bottom-most cube of (<ref>) by constructing two hypercubes. The first hypercube is obtained by pairing the back face of (<ref>) with the hypercube

\[
\begin{tikzcd}[column sep=2cm]
\as \bar \bs \ar[r, "\Theta_{\a \bar \b, \a'\bar \b'}"]& \as'\bar \bs'
\end{tikzcd}.
\]

The second hypercube is obtained by pairing the entirety of  (<ref>) with the 0-dimensional hypercube $\as' \bar \bs'$.

We now prove the existence of the hypercube in (<ref>).  The existence of length 2 chains in Equation (<ref>) along the top, left and right faces is straightforward.

We now consider the bottom face of (<ref>). The desired relation is

\[
A_\tau(\Theta_{\a \bar \a \bar \g_0, \a'\bar \a' \g_0\bar \g_0})+B_\tau(\Theta_{\a \bar \a \bar \g_0, \a'\bar \a' \g_0\bar \g_0})=\d (h)
\]

where $h$ denotes the diagonal chain. Since $\tau$ is a closed curve, we have an equality $A_\tau=B_{\tau}$, the above equation is satisfied by $h=0$. (Compare Equation (<ref>) for arcs instead of closed curves).
As well will see, setting $h=0$ might not allow the length 3 hypercube relation to be satisfied, so instead we use the chain

\begin{equation}
h=\veps\cdot \Theta_{\a\bar \a \g_0\bar \g_0,\a'\bar \a'\g_0\bar \g_0},
\label{eq:length-2-map-1-handle-T}
\end{equation}

where $\veps\in \bF$ is to be determined.

Finally, we claim that we may make a choice of the constant $\veps$ in equation (<ref>), and also pick a length 3 chain for the cube in (<ref>) so that the hypercube relations are satisfied.  To see this, let $C\in \CF(\Ds \g_0 \bar \g_0, \as'\bar \as' \g_0 \bar \g_0)$ denote the sum of all compositions featuring in the length 3 hypercube relations, except for the length 3 map itself. Here, we momentarily set $\veps=0$ in (<ref>). The hypercube relations on the proper faces force $C$ to be a cycle. We note however that $C$ has the same grading as the top degree element generator of $\HF^-(\Ds\g_0\bar \g_0, \as'\bar \as' \g_0\bar \g_0)$. In particular, it either represents the top degree element or is null-homologous. If it is null-homologous, we set $\veps=0$ in (<ref>) and pick any primitive of $C$ in the appropriate grading to be the length 3 map in our hypercube. If $C$ represents the top degree generator, then we pick $\veps=1$, and let the length 3 chain be any primitive of

\[
f_{\Dt\g_0\bar \g_0, \a \bar \a \g_0\bar \g_0, \a'\bar \a' \g_0\bar \g_0}(\Theta_{\Dt\g_0\bar \g_0, \a \bar \a \g_0\bar \g_0},\Theta_{\a \bar \a \g_0\bar \g_0, \a'\bar \a' \g_0\bar \g_0})+C.
\]

The hypercube relations are clearly satisfied.

Finally, it remains to extend the bottom-most cube of Figure <ref>. To extend this hypercube, it is convenient to consider the expanded description in (<ref>). In fact, the only hypercube which requires explanation to extend is the cube

\[
\begin{tikzcd}[
row sep=1cm,
column sep=1cm,
fill opacity=.7,
text opacity=1
\CF(\as \bar \bs \bar \g_0, \as \bar \as \bar \g_0)
\ar[r, "F_1^{\g_0, \g_0}"]
\ar[d,equal]
\CF(\as \bar \bs \g_0\bar \g_0, \as \bar \as \g_0\bar \g_0)
\ar[r, "B_\tau"]
\CF(\as \bar \bs \g_0 \bar \g_0, \as \bar \as \g_0\bar \g_0)
\ar[d, "F_3^{\g_0,\g_0}"]
\\
\CF(\as \bar \bs \bar \g_0, \as \bar \as \bar \g_0)
\ar[rr, equal]
\CF(\as \bar \bs \bar \g_0, \as \bar \as \bar \g_0)
\end{tikzcd}
\]

We define our 3-dimensional extension of this cube to have trivial length 3 map. The hypercube relations for the 3-dimensional extension of this cube are proven somewhat similarly to the proof of Proposition <ref>. The only non-trivial length 2 map in the above diagram is in the top subface. This subface has the following form

\[
\begin{tikzcd}[labels=description,column sep=1.3cm, row sep=1cm]
\CF(\as \bar \bs \bar \g_0,\as \bar \as\bar \g_0)
\ar[r, "F_1^{\g_0,\g_0}"]
\ar[d]
\CF(\as \bar \bs\g_0 \bar \g_0, \as \bar \as\g_0\bar \g_0)
\ar[r, "B_\tau"]
\ar[d]
\ar[dr,dashed]&
\CF(\as \bar \bs\g_0 \bar \g_0, \as \bar \as\g_0\bar \g_0)
\ar[d]\\
\CF(\as' \bar \bs' \bar \g_0,\as \bar \as\bar \g_0)
\ar[r, "F_1^{\g_0,\g_0}"]
\ar[d]
&\CF(\as' \bar \bs'\g_0 \bar \g_0, \as \bar \as\g_0\bar \g_0)
\ar[r, "B_{\tau}"]
\ar[d]
\ar[dr,dashed]
\CF(\as' \bar \bs'\g_0 \bar \g_0, \as \bar \as\g_0\bar \g_0)
\ar[d]\\
\CF(\as' \bar \bs' \bar \g_0,\as' \bar \as'\bar \g_0)
\ar[r, "F_1^{\g_0,\g_0}"]&
\CF(\as' \bar \bs'\g_0 \bar \g_0, \as' \bar \as'\g_0\bar \g_0)
\ar[r, "B_{\tau}"]&
\CF(\as' \bar \bs'\g_0 \bar \g_0, \as' \bar \as'\g_0\bar \g_0)
\end{tikzcd}
\]

It is sufficient to show that the compression of this diagram composes trivially with the map $F_3^{\g_0, \g_0}$. The key tool we will use is that the triangle maps, when applied to elements of the form $(\xs\times\theta_{\g_0,\g_0}^+)\otimes (\ys\times \theta_{\g_0,\g_0}^+)$ have only summands of the form $\zs\times \theta_{\g_0,\g_0}^+$. This fact follows from Proposition <ref>. The top diagonal map counts holomorphic triangles which are weighted by the quantity $\# (\d_{\a\bar \a \g_0\bar \g_0}(\psi)\cap \tau)\in \bF$. Similarly, the bottom triangle map is a sum of holomorphic triangle maps with certain weights. The special inputs for the length 1 triangle maps all have tensor factors of $\theta_{\g_0,\g_0}^+$, as does the special length 2 element used for the bottom diagonal map by equation (<ref>). Hence, in compression of the diagram, the only holomorphic triangles which are counted have only tensor factors of $\theta_{\g_0,\g_0}^+$, and hence so does their output. In particular, the output vanishes when composed with $F_3^{\g_0,\g_0}$.

§.§ 3-handles

We define the 3-handle map by dualizing the construction of the 1-handle map. Concretely, this amounts to compressing the diagram shown in Equation (<ref>).

\begin{equation}
\begin{tikzcd}[
row sep=1cm,
column sep=2.6cm,
fill opacity=.7,
text opacity=1
\CF(\as\g_0,\bs\g_0)
\ar[d, "F_{1}^{\bar{\b}\bar \g_0,\bar{\b}\bar \g_0}"]
\ar[rr, "F_3^{\g_0,\g_0}"]
\CF(\as,\bs)
\ar[d,"F_{1}^{\bar{\b},\bar{\b}}"]
\\
\CF(\as \bar{\bs} \g_0\bar \g_0, \bs \bar{\bs} \g_0\bar \g_0)
\ar[r, "B_\tau"]
\ar[d,"f^{\b \bar{\b} \g_0\bar \g_0\to \Dt\Dt_0}_{\a \bar{\b}\g_0 \bar \g_0}"]
\ar[dr,dashed]
\CF(\as \bar \bs \g_0 \bar \g_0, \bs  \bar \bs \g_0  \bar \g_0)
\ar[r,"F_3^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[d, "f^{\b \bar \b\g_0 \bar \g_0\to \Dt \g_0 \bar \g_0}_{\a \bar \b \g_0 \bar \g_0}"]
& \CF(\as\bar \bs, \bs \bar \bs)
\ar[d,"f^{\b \bar \b \to \Dt }_{\a \bar \b }"]
\\
\CF(\as \bar \bs \g_0 \bar \g_0,\Ds \Ds_0)
\ar[r, "f_{\a \bar \g_0\bar \g_0}^{\Dt\Dt_0\to \Dt \g_0\bar \g_0}"]
\ar[d, "f^{\Dt\to \a  \bar{\a}}_{\a  \bar{\b}}"]
\ar[dr,dashed]
\CF(\as \bar \bs  \g_0 \bar \g_0, \Ds \g_0 \bar \g_0)
\ar[r, "F_3^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[d, "f_{\a \bar \b \g_0 \bar \g_0}^{\Dt \g_0 \bar \g_0\to \a \bar \a \g_0 \bar \g_0}"]
& \CF(\as\bar{\bs},\Ds)
\ar[d, "f_{\a \bar \b}^{\Dt\to \a \bar \a}"]
\\
\CF(\as  \bar{\bs} \g_0\bar \g_0, \as  \bar{\as} \g_0\bar \g_0)
\ar[r, equal]
\ar[d, "F_3^{\a \g_0,\a \g_0}"]
\CF(\as \bar \bs \g_0 \bar \g_0, \as \bar \as \g_0 \bar \g_0)
\ar[r, "F_3^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\CF(\as \bar \bs, \as \bar \as)
\ar[d, "F_3^{\a,\a}"]
\\
\CF(\bar{\bs}\bar \g_0, \bar{\as} \bar \g_0)
\ar[rr, "F_3^{\bar \g_0, \bar \g_0}"]
\CF(\bar{\bs},\bar{\as})
\end{tikzcd}
\label{eq:3-handles}
\end{equation}

§.§ Attaching 1-handles near the basepoint

We may obtain a slightly simpler formula for the 1-handle map when we perform the attachment near the basepoint $w$.

Suppose $\bS_0\subset Y$ is a 0-sphere and $Y$ is connected. We have an isomorphism of groups

\[
\CFI^-(Y(\bS_0))\iso \CFI^-(Y)\otimes_{\bF} \langle \theta^+,\theta^-\rangle.
\]

With respect to this decomposition, the involutive 1-handle map takes the form

\[
\CFI(W(\bS_0))(\xs)=\xs\times \theta^+,
\]

extended equivariantly over $\bF[U,Q]/Q^2$. The 3-handle map takes the form $\xs\times \theta^+\mapsto 0$ and $\xs\times \theta^-\mapsto \xs$, extended $\bF[U,Q]/Q^2$-equivariantly.

We focus on the 1-handle map. We pick our diagram so that, before doubling, there is a special genus 1 region near the basepoint, and so the doubled arcs have the configuration shown in Figure <ref> in the double of this region. We form the compression of (<ref>) by first compression horizontally, then vertically.

We first show that for suitable almost complex structures, the diagonal maps make trivial contribution to compression of the diagram. We consider the top-most diagonal map. This map counts holomorphic quadrilaterals on the connected sum

\[
(\as \bar \bs, \bs \bar \bs, \Ds, \Ds)\# (\g_0\bar \g_0,\g_0\bar \g_0,\g_0\bar \g_0,\Ds_0).
\]

Note that each Heegaard subdiagram of the right-handle quadruple has vanishing differential (since it has the minimal number of generators possible; see Figure <ref>). Hence the stabilization results for holomorphic quadrilaterals from Proposition <ref> imply that

\[
h_{\a \bar \b \g_0\bar \g_0}^{\b \bar \b \g_0\bar \g_0\to \Dt \g_0\bar \g_0\to \Dt \Dt_0}\circ F_1^{\g_0 \bar \g_0, \g_0\bar \g_0}=F_1^{\g_0\bar \g_0, \Dt_0} \circ h_{\a\bar \b}^{\b \bar \b \to \Dt\to \Dt}.
\]

By the small translate theorem for holomorphic quadrilaterals <cit.>, the map $h_{\a\bar \b}^{\b \bar \b \to \Dt\to \Dt}$ vanishes.

We now consider the bottom-most diagonal map in (<ref>). Because of our choice of diagram, the chain $\eta$ in (<ref>) may be chosen to vanish, by the same model computation we used to verify Lemma <ref>.  Hence, the dashed length 2 map is the sum

\begin{equation}
h^{\Dt \g_0\bar \g_0\to \Dt \Dt_0\to \a \bar \a \g_0\bar \g_0}_{\a \bar \b \g_0\bar \g_0}
+h_{\a \bar \b \g_0\bar \g_0}^{\Dt \g_0\bar \g_0\to \a \bar \a \g_0\bar \g_0\xrightarrow{\tau} \a \bar \a \g_0\bar \g_0}
\label{eq:holomorphic-quadrilaterals-1-handle-diagonal}
\end{equation}

Here, $h_{\a \bar \b \g_0\bar \g_0}^{\Dt \g_0\bar \g_0\to \a \bar \a \g_0\bar \g_0\xrightarrow{\tau} \a \bar \a \g_0\bar \g_0}$ denotes the map which counts holomorphic triangles on the diagram $(\Sigma'\# \bar \Sigma', \as \bar \bs \g_0, \bar \g_0, \Ds \g_0\bar \g_0, \as \bar \as \g_0\bar \g_0)$ which are weighted by an extra factor of $\d_{\a \bar \a \g_0\bar \g_0} D(\psi)\cdot \tau$.

We claim that both of these maps in (<ref>) vanish when precomposed with $F_1^{\g_0\bar \g_0,\g_0\bar \g_0}$ and post-composed with $F_3^{\a \g_0,\a \g_0}$. Consider the first quadrilateral map. It counts curves on the diagram

\[
(\as \bar \bs, \Ds, \Ds, \as \bar \as)\# (\g_0\bar \g_0, \g_0\bar \g_0, \Ds_0, \g_0\bar \g_0 ).
\]

We can use Proposition <ref> to destabilize the quadruple, noting that the evaluation of the right-hand triple on a degree 0 tree of $C_*^{\cell}(K_4)$ is $\theta^+|\bar \theta^-+\theta^-|\bar \theta^+$ by Lemma <ref>. Hence $\theta_{\g_0,\g_0}^-$ factor of the composition is

\[
h_{\a \bar \b}^{ \Dt \to \Dt\to \a \bar \a}(\xs)\otimes \theta_{\bar \g_0, \bar \g_0}^+.
\]

This expression vanishes by the small translate theorem for quadrilaterals <cit.>, since $\Ds$ appears twice.

Finally, we argue that the triangle counting map $h_{\a \bar \b \g_0\bar \g_0}^{\Dt \g_0\bar \g_0\to \a \bar \a \g_0\bar \g_0\xrightarrow{\tau} \a \bar \a \g_0\bar \g_0}$ also makes trivial contribution. In this case, the triangle count occurs on

\[
(\as \bar \bs, \Ds, \as \bar \as)\#(\g_0\bar \g_0, \g_0\bar \g_0, \g_0\bar \g_0).
\]

Proposition <ref> implies there are no triangles with $\theta_{\g_0,\g_0}^-$ as output when only terms involving $\theta_{\g_0,\g_0}^+$ are the input, so this triangle map makes trivial contribution to the compressed diagram.

The argument for 3-handles follows because the 3-handle map is defined dually to the 1-handle map, so the above argument immediately translates after taking duals.

§.§ Commutation of 1-handles

Suppose that $\bS$ and $\bS'$ are two embedded and disjoint spheres in $Y$, whose dimensions lie in $\{0,2\}$. Then

\[
\CFI(W(Y(\bS),\bS'))\circ \CFI(W(Y,\bS))\simeq \CFI(W(Y(\bS'),\bS))\circ
\CFI(W(Y,\bS')).
\]

Lemma <ref> may be proven by using the model computations from Lemma <ref>. Alternatively, it may be proven by building a 3-dimensional hyperbox which has two faces corresponding to the definition of the 1-handle maps from (<ref>).

§ 2-HANDLES

In this section, we describe the map for a framed link $\bL\subset Y$, with a self-conjugate $\Spin^c$ structure $\frs$ on $W(Y,\bL)$ (or more generally, a family $\frS\subset \Spin^c(W(Y,\bL))$ which is closed under conjugation). As in Section <ref>, we may omit the path and framing from our notation.

The existence of a map for 2-handles was described by Hendricks and Manolescu <cit.>, though they were not able to prove invariance of the construction. The construction we present in this section should be thought of as a systematic version of the construction therein using the doubling operation.

§.§ The construction

Suppose $\bL$ is a framed link in $Y$. We assume that the framing is Morse, so that there is a corresponding 2-handle cobordism $W(Y,\bL)$. We pick a bouquet for $\bL$, in the sense of <cit.>*Definition 4.1, as well as a Heegaard triple $(\Sigma,\as',\as,\bs)$ subordinate to this bouquet <cit.>*Definition 4.2.

Like the 2-handle maps from <cit.>, our 2-handle maps also depend on a bouquet for the link $\bL$. Independence from the bouquet will be proven in Lemma <ref>.

We first construct several auxiliary hypercubes. We firstly have two hypercubes of alpha attaching circles

\begin{equation}
\begin{tikzcd}[column sep=1.7cm]
\as \ar[r, "\Theta_{\a', \a}"] & \as'
\end{tikzcd}
\quad
\text{and}
\quad
\begin{tikzcd}[column sep=1.7cm]
\as \bar \bs \ar[r, "\Theta_{\a' \bar \b, \a \bar \b}"] & \as'\bar \bs.
\end{tikzcd}
\label{eq:easy-aux-hypercubes-2-handles}
\end{equation}

and a hypercube of beta attaching circles

\begin{equation}
\begin{tikzcd}[column sep=1.7cm]
\bs\bar \bs \ar[r, "\Theta_{\b\bar \b, \Dt}"]& \Ds
\end{tikzcd}
\label{eq:easy-aux-hypercubes-2-handles-beta}
\end{equation}

There is a unique choice of top degree cycle in the hypercubes in Equation (<ref>), while the cycle $\Theta_{\b\bar \b,\Dt}$ in Equation (<ref>) is only well-defined up to addition of a boundary.

Next we construct a more complicated hypercube of beta attaching curves:

\begin{equation}
\begin{tikzcd}[
row sep=1cm,
column sep=3cm,
fill opacity=.7,
text opacity=1
\Ds
\ar[r, "\Theta_{\Dt,\a'\bar\a'}"]
\ar[d, "\Theta_{\Dt, \a\bar \a}"]
\ar[dr,dashed, "\lambda_{\Dt,\a\bar\a'}"]
\as'\bar \as'
\ar[d, "\Theta_{\a'\bar\a', \a \bar \a'}"]
\\
\as \bar \as
\ar[r, "\Theta_{\a \bar \a,\a \bar \a'}"]
\as \bar \as',
\end{tikzcd}
\label{eq:second-auxiliary-hypercube-2-handle}
\end{equation}

We now prove that the hypercube in (<ref>) may be constructed:
If $\Theta_{\Dt,\a'\bar \a'}$, $\Theta_{\a'\bar \a', \a \bar \a'}$, $\Theta_{\Dt,\a \bar \a}$ and $\Theta_{\a \bar \a, \a \bar \a'}$ are cycles which represent the top degree elements of their respective $\HF^-$ groups, then

\[
\left[f_{\Dt, \a'\bar \a', \a \bar \a'}(\Theta_{\Dt, \a'\bar \a'}, \Theta_{\a'\bar \a', \a \bar \a'})\right]=\left[f_{\Dt, \a \bar \a, \a \bar \a'}(\Theta_{\Dt, \a \bar \a},\Theta_{\a \bar \a, \a \bar \a'})\right],
\]

where brackets denote the induced element in homology.
In particular, $\lambda_{\Dt,\a \bar \a'}\in \CF(\Ds,\as\bar \as')$ may be chosen so that (<ref>) is a hypercube of attaching curves.

Both triples $(\Sigma\# \bar \Sigma,\Ds, \as'\bar \as', \as \bar \as')$ and $(\Sigma\# \bar \Sigma,\Ds, \as\bar \as, \as \bar \as')$ represent the same cobordism, topologically. Namely, they represent a cobordism from $(S^1\times S^2)^{\# g}$ to $(S^1\times S^2)^{g-|\bL|}$ which is surgery on a $|\bL|$ component link, each component of which is an $S^1$-fiber of a $S^1\times S^2$-summand.

Since $\left[f_{\Dt, \a'\bar \a', \a \bar \a'}(\Theta_{\Dt, \a'\bar \a'}, \Theta_{\a'\bar \a', \a \bar \a'})\right]$ represents a top-degree class of $\CF(S^1\times S^2)^{g-|\bL|}$, we have that any two choices of $\lambda_{\Dt,\a\bar\a'}$ satisfying (<ref>) are homologous.

Now suppose that $\frS\subset \Spin^c(W(Y,\bL))$ is a finite set which is closed under conjugation. We now define the 2-handle map for $\bL$, which we denote by $\CFI(W(Y,\bL),\frS)$, using the hypercubes in (<ref>),  (<ref>) and (<ref>). We define the map $\CFI(W(Y,\bL),\frS)$ to be the compression of the hyperbox shown in equation (<ref>).  A priori, $\CFI(W(Y,\bL), \frS)$ depends on the choices in its construction; for well-definedness see Section <ref>.

\begin{equation}
\begin{tikzcd}[
row sep=1cm,
column sep=3cm,
fill opacity=.7,
text opacity=1
\CF(\as,\bs)
\ar[rr, "f_{\a\to \a'}^{\b}"]
\ar[d, "F_1^{\bar \b, \bar \b}"]
\CF(\as', \bs)
\ar[d, "F_1^{\bar \b, \bar \b}"]
\\
\CF(\as\bar\bs, \bs \bar \bs)
\ar[rr, "f_{\a \bar \b\to \a' \bar \b}^{\b \bar \b}"]
\ar[d,"f_{\a \bar \b}^{\b \bar \b\to \Dt}"]
\ar[drr,dashed, "h_{\a\bar \b\to \a'\bar \b}^{\b\bar \b\to \Dt}"]
\CF(\as '\bar \bs, \bs \bar \bs)
\ar[d,"f_{\a' \bar \b}^{\b \bar \b\to \Dt}"]
\\
\CF(\as\bar \bs, \Ds)
\ar[rr,"f_{\a\bar \b\to \a'\bar \b}^{\Dt}"]
\ar[d,equal]
\ar[drr,dashed," h_{\a\bar\b\to \a'\bar \b}^{\Dt\to \a'\bar \a'}"]
\CF(\as' \bar \bs, \Ds)
\ar[d,"f_{\a'\b}^{\Dt\to \a'\bar \a'}"]
\\
\CF(\as \bar \bs, \Ds)
\ar[d,"f_{\a \bar{\b}}^{\Dt\to \a \bar \a}"]
\ar[dr,dashed, " H_{\a\bar \b}^{\Dt\to \a \bar\a'} "]
\ar[r, "f_{\a \bar \b}^{\Dt\to \a'\bar \a'} "]
\CF(\as \bar \bs, \as'\bar \as')
\ar[r, "f_{\a \bar \b\to \a'\bar \b}^{\a'\bar \a'}"]
\ar[d, "f_{\a \bar \b}^{\a' \bar \a'\to \a \bar \a'}"]
\CF(\as' \bar \bs, \as' \bar \as')
\ar[d, "F_3^{\a',\a'}"]
\\
\CF(\as \bar \bs, \as \bar \as)
\ar[r, "f_{\a \bar \b}^{\a\bar \a\to \a \bar \a'}"]
\ar[d,"F_3^{\a,\a}"]
\CF(\as \bar \bs, \as \bar \as')
\ar[r, "F_3^{\a, \a}"]
\ar[d, "F_3^{\a,\a}"]
\CF( \bar \bs, \bar \as')
\ar[d,equal]
\\
\CF(\bar \bs, \bar \as)
\ar[r,"f_{\bar \b}^{\bar \a\to \bar \a'}"]
\CF(\bar \bs, \bar \as')
\ar[r,equal]
\CF(\bar \bs, \bar \as')
\end{tikzcd}
\label{eq:hyperbox-2-handles}
\end{equation}

If we compress each horizontal level in (<ref>), then the resulting diagram is a hyperbox of chain complexes.

It suffices to show that each face is a hypercube of chain complexes. The subcubes with length 2 maps are all obtained by pairing hypercubes of attaching curves, so the claim is immediate for these faces. The top-most and bottom-most subcubes are hypercubes of stabilization, so the claim is immediate. The only remaining cube of interest is the second-bottom-most cube on the right. The hypercube relations amount to the relation

\begin{equation}
F_{3}^{\a',\a'}\circ f_{\a \bar \b\to \a'\bar \b}^{\a' \bar \a'}=F_3^{\a,\a}\circ f_{\a \bar \b}^{\a'\bar \a'\to \a \bar \a'}.
\label{eq:2-handle-map-relation-with-3-handles}
\end{equation}

This is proven as follows. The Heegaard triples for these triangle maps are $(\Sigma, \as',\as,\as')\#(\bar \Sigma, \bar \bs, \bar \bs, \bar \as')$ and $(\Sigma, \as, \as', \as)\#(\bar \Sigma, \bar \bs, \bar \as', \bar \as')$. In particular, both triples are stabilizations of a Heegaard triple for a small isotopy. Using the stabilization results of Proposition <ref> and the small triangle theorem <cit.>*Proposition 11.1, we obtain  (<ref>), completing the proof.

If $\frS\subset \Spin^c(W(Y,\bL))$ is a set of $\Spin^c$ structures which is closed under the conjugation action, then a cobordism map may be defined by summing the maps above for all $\frs\in \frS$. If $\frS$ is infinite, one may need to use coefficients in $\bF[[U]]$ to obtain a well defined map.

§.§ Independence from simple handleslides

We now prove that the construction above commutes with the maps for simple handleslides and changes of the doubling arcs. We phrase our result in enough generality that they also imply that the 2-handle maps are independent of the choice of bouquet for $\bL$, and are independent from handleslides amongst the components of $\bL$.

Suppose that $(\Sigma,\as',\as,\bs)$ is a Heegaard triple subordinate to a bouquet of $\bL\subset Y$. Suppose that $(\Sigma,\as'_H,\as_H,\bs_H)$ is another Heegaard triple, such that $\as'_H$, $\as_H$ and $\bs_H$ are obtained from $\as',$ $\as$ and $\bs$ (respectively) by elementary handleslide equivalences, in the sense of Section <ref>. We assume, furthermore that each of the following sets admits a unique top degree intersection point, which is a cycle in its respective Floer complex:

\[
\bT_{\a_H}\cap \bT_{\a}, \quad \bT_{\a_H'}\cap \bT_{\a'}, \quad \bT_{\a_H'}\cap \bT_{\a}, \quad \text{and} \quad \bT_{\b}\cap \bT_{\b_H}.
\]

We also suppose that we pick two sets of doubling arcs, $\Ds$ and $\Ds_H$. We require no particular relation between $\Ds$ and $\Ds_H$, except that the diagram $(\Sigma\# \bar \Sigma,\Ds,\Ds_H,w)$ be admissible.

Suppose that $(\Sigma,\as',\as,\bs)$ and $(\Sigma,\as'_H,\as_H,\bs_H)$ are as above.
Let $\CFI(W(\bL))$ and $\CFI(W(\bL))^H$ denote the 2-handle maps, computed with the triples $(\Sigma,\as',\as,\bs)$ and $(\Sigma,\as'_H,\as_H,\bs_H)$, and the doubling arcs $\Ds$ and $\Ds_H$, respectively. Let $\frD$ denote the doubling data for $Y$ determined by $(\Sigma,\as,\bs)$, $\Ds$, and some choices of almost-complex structures. Let $\frD_H$ denote the doubling data determined by $(\Sigma,\as_H,\bs_H)$, $\Ds_H$, and some choices of almost-complex structures. Let $\frD'$ and $\frD_H'$ denote the analogous data for $Y(\bL)$. Then

\[
\Psi_{\frD'\to \frD_H'} \circ \CFI(W(\bL))\simeq \CFI(W(\bL))^H\circ \Psi_{\frD\to \frD_H}.
\]

Here, $\Psi_{\frD\to \frD_H}$ and $\Psi_{\frD'\to \frD'_H}$ denote the transition maps for elementary handleslides and changes of the doubling data, and $\simeq$ denotes $\bF[U,Q]/Q^2$-equivariant chain homotopy.

Mirroring the construction of the maps $\Psi_{\frD\to \frD_H}$, we construct for each subcube of Equation (<ref>) a hyperbox of size $(2,1,1)$. The final coordinate corresponds to the involution. If we collapse this axis of the cube, we obtain a hyperbox of size $(2,1)$. If we compress this hyperbox, we obtain the following hypercube, which realizes the homotopy commutation in the statement.

\[
\begin{tikzcd}[
row sep=1cm,
column sep=3cm,
fill opacity=.7,
text opacity=1
\CFI(\frD)
\ar[dr, "\Psi_{\frD\to \frD_H}"]
\ar[r,"F_{W(\bL)}"]
\ar[drr,dashed]
\CFI(\frD')
\ar[dr,  "\Psi_{\frD'\to \frD'_H}"]
\\
\ar[r,"F_{W(\bL)}^H"]
\CFI(\frD'_H)
\end{tikzcd}
\]

The top-most hyperbox is obtained by extending the top-most hypercube of Equation (<ref>) out of the page, and is obtained by stacking two hypercubes of stabilization. We leave the details to the reader.

We now consider the hypercube in (<ref>) which is second from the top. The corresponding hyperbox is obtained by pairing hypercubes, as follow. We think of a hyperbox of size $(2,1,1)$ as being obtained by stacking two hypercubes. The first hypercube is obtained by constructing and pairing the following hypercubes of attaching curves, where the length-$1$ edges are obtained using the unique top-degree generators specified earlier; the $2$-faces are filled as in Lemma <ref>:

\[
\begin{tikzcd}[
row sep=1cm,
column sep=2cm,
fill opacity=.7,
text opacity=1
\as \bar \bs
\ar[r]
\ar[drr,dashed]
\ar[dr]
\as'\bar \bs
\ar[dr]
\,
\\
\as_H \bar \bs_H
\ar[r]
\as_H' \bar \bs_H
\end{tikzcd}
\quad \text{and} \quad
\begin{tikzcd}[
row sep=1cm,
column sep=3cm,
fill opacity=.7,
text opacity=1
\bs \bar \bs\ar[d]\\
\Ds
\end{tikzcd}.
\]

The next hypercube is obtained by constructing and pairing the following hypercubes of attaching curves:

\[
\begin{tikzcd}[
row sep=1cm,
column sep=2cm,
fill opacity=.7,
text opacity=1
\as_H\bar \bs_H \ar[r]
\as_H'\bar \bs_H
\end{tikzcd}
\quad
\text{and}
\quad
\begin{tikzcd}[
row sep=1cm,
column sep=1.2cm,
fill opacity=.7,
text opacity=1
\bs \bar \bs
\ar[d]
\ar[dr]
\ar[ddr,dashed]
\Ds
\ar[dr]
& \bs_H \bar \bs_H
\ar[d]
\\
& \Ds_H
\end{tikzcd}
\]

The remaining hypercubes which have a length 2 map are constructed by pairing hypercubes of attaching curves in a straightforward manner. We remark that the central-most hypercube of (<ref>) is obtained by a small rearrangement of the pairing of an alpha hypercube with a beta hypercube. The corresponding 3-dimensional hyperbox is obtained by performing the analogous rearrangement to two hypercubes obtained by pairing.

Finally, we consider the three bottom-most hypercubes of (<ref>) which do not have a length-$2$ map. We consider the right hypercube in the second to bottom-most level.

\begin{equation}
\begin{tikzcd}[
column sep={1.6cm,between origins},
row sep=1cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;
\CF(\as \bar \bs, \as'\bar \as')
\ar[dr]
\ar[ddd]
\ar[rr]
&[.7 cm]
&\CF(\as' \bar \bs, \as'\bar \as')
\ar[dr]
\ar[ddd, "F_3^{\a',\a'}"]
&[.7 cm]
\\
&[.7 cm]
|[alias=A]|\CF(\as_H \bar \bs_H, \as'\bar \as')
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr]
&[.7 cm]
\CF (\as'_H \bar \bs_H, \as'\bar \as')
\ar[ddd, "F_3^{\a_H',\a'}"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed]
\\
\\
\CF(\as \bar \bs, \as \bar \as')
\ar[dr]
\ar[rr, "F_3^{\a,\a}"]
&[.7 cm]\,&
\CF(\bar \bs, \bar \as)
\ar[dr]
\\
& [.7 cm]
\CF(\as_H \bar \bs_H, \as \bar \as')
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu]
\ar[rr, "F_3^{\a_H,\a}"]
\ar[from=uuuul,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuul,dashed]
&[.7 cm]
\CF(\bar \bs_H, \bar \as')
\end{tikzcd}
\hspace{-1.2cm}
\begin{tikzcd}[
column sep={1.6cm,between origins},
row sep=1cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;
\CF(\as_H \bar \bs_H, \as'\bar \as')
\ar[dr]
\ar[ddd]
\ar[rr]
&[.7 cm]
\CF(\as'_H \bar \bs_H, \as'\bar \as')
\ar[rd]
\ar[ddd, "F_3^{\a_H',\a'}"]
&[.7 cm]
\\
&[.7 cm]
\CF(\as_H\bar \bs_H, \as'_H \bar \as'_H)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr]
&[.7 cm]
\CF (\as_H' \bar \bs_H, \as'_H\bar \as'_H)
\ar[ddd, "F_3^{\a_H',\a_H'}"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed]
\\
\\
\CF(\as_H \bar \bs_H, \as \bar \as')
\ar[dr]
\ar[rr, "F_3^{\a_H, \a}"]
&[.7 cm]&
\CF(\bar \bs_H, \bar \as')
\ar[dr]
\\
& [.7 cm]
\CF(\as_H \bar \bs_H, \as_H \bar \as_H')
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu]
\ar[rr, "F_3^{\a_H,\a_H}"]
\ar[from=uuuul,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuul,dashed]
&[.7 cm]
\CF(\bar \bs_H, \bar \as_H')
\end{tikzcd}
\label{eq:hyperbox-2-handle-bottom-right}
\end{equation}

In (<ref>), all of the length 1 and 2 maps which are unlabeled are holomorphic triangle maps or quadrilateral maps. There are no length 3 maps.

The length 2 hypercube relations for the diagrams in (<ref>) are verified as follows. For the faces which have length 2 maps, the hypercube relations are obvious. For the other faces, the length 2 relations are proven similarly to the proof of (<ref>).

It remains to prove the length 3 relations for the diagrams in  (<ref>). To verify these, we note that each summand in the length 3 relation is a composition of a sum of quadrilateral maps followed by a  3-handle map. We claim that each summand vanishes identically. To see this, consider the length 2 map along the top face of the left cube of (<ref>). This map is the sum of two quadrilateral maps, which take place on the diagrams $(\Sigma, \as'_H,\as',\as,\as')\#(\bar \Sigma,\bar \bs_H, \bar \bs, \bar \bs, \bar \as')$ and $(\Sigma, \as'_H,\as_H,\as,\as')\#(\bar \Sigma,\bar \bs_H, \bar \bs_H, \bar \bs, \bar \as') $. In particular, both diagrams are algebraically rigid and admissible multi-stabilizations. By our result on stabilizations in Proposition <ref>, we see that

\[
F_3^{\a_H',\a_H}\circ h_{\a \bar \b\to \a'\bar \b\to \a_H'\bar \b_H}^{\a' \bar \a'}=h_{\bar \b \to \bar \b\to \bar \b_H}^{\bar \a'}\circ F_3^{\a, \a'},
\]

however $h_{\bar \b \to \bar \b\to \bar \b_H}^{\bar \a'}=0$ by the small translate theorem for quadrilaterals <cit.>*Proposition 11.5, since $\bar \bs$ is repeated twice in the diagram. The same argument works for all of the other summands, verifying the hypercube relations for the diagrams in (<ref>).

One now constructs hypercubes for the bottom two faces of (<ref>). The construction of these hyperboxes is similar to the ones constructed above, and we leave the details to the reader.

There is one final hyperbox to construct, corresponding to the lowest level of Figure <ref>. This hyperbox is constructed easily by pairing hypercubes of attaching curves and rearranging the result slightly. We leave the details to the reader.

§.§ The composition law

In this section, we prove the composition law for cobordisms with only 2-handles.

Suppose that $\bL_1$ and $\bL_2$ are two framed links in $Y$, and $\frs_1\in \Spin^c(W(Y,\bL_1))$ and $\frs_2\in \Spin^c(W(Y(\bL_1),\bL_2))$ are two self-conjugate $\Spin^c$ structures. Then

\[
\CFI(W(Y(\bL_1),\bL_2),\frs_2)\circ \CFI(W(Y,\bL_1),\frs_1)\simeq \CFI(W(Y,\bL_1\cup \bL_2),\frS(\frs_1,\frs_2))
\]

where $\frS(\frs_1,\frs_2)$ is the set of $\Spin^c$ structures on $W(Y,\bL_1\cup \bL_2)$ which restrict to $\frs_1$ and $\frs_2$.

The proof of Proposition <ref> is to construct a 3-dimensional hypercube with the following properties:

* One direction corresponds to the involution.

* One face corresponds to the cobordism map $\CFI(W(Y,\bL_1),\frs_1)$.

* One face corresponds to the cobordism map $\CFI(W(Y(\bL_1),\bL_2), \frs_2)$.

* One face corresponds to the identity map from $\CFI(Y)$ to itself.

* One face corresponds to the map $\CFI(W(Y,\bL_1\cup \bL_2), \frS(\frs_1,\frs_2))$.

If we view the 3-dimensional hypercube as forming a 2-dimensional hypercube involving the $\CFI$ complexes, then the corresponding shape is as follows:

\[
\begin{tikzcd}[
row sep=1cm,
column sep=3cm,
fill opacity=.7,
text opacity=1
\CFI(Y)
\ar[r, "F_{W(Y,\bL_1),\frs_1}"]
\ar[dr,"F_{W(Y,\bL_1\cup \bL_2),\frS(\frs_1,\frs_2)}"]
\ar[drr,dashed]
\CFI(Y(\bL_1))
\ar[dr, "F_{W(Y(\bL_1),\bL_2),\frs_2}"]
\\
\ar[r, "\id"]
\CFI(Y(\bL_1,\bL_2))
\end{tikzcd}
\]

As a first step, pick a Heegaard quadruple $(\Sigma,\as'',\as',\as,\bs)$ satisfying the following:

* $(\Sigma,\as',\as,\bs)$ is subordinate to a bouquet of the link $\bL_1\subset Y$.

* $(\Sigma,\as'',\as',\bs)$ is subordinate to a bouquet of the link $\bL_2\subset Y(\bL_1)$.

* $(\Sigma,\as'',\as,\bs)$ is subordinate to a bouquet of the link $\bL_1\cup \bL_2\subset Y$.

Let $\Theta_{\a',\a}$, $\Theta_{\a'',\a'}$ and $\Theta_{\a'',\a}$ be the canonical generators. As in Ozsváth and Szabó's original proof for $\HF^-$, an easy model computation implies that the following diagram is a hypercube of alpha attaching curves:

\begin{equation}
\begin{tikzcd}[
row sep=1cm,
column sep=3cm,
fill opacity=.7,
text opacity=1
\as
\ar[r, "\Theta_{\a',\a}"]
\ar[d, "\Theta_{\a'', \a}"]
\as'
\ar[d, "\Theta_{\a'',\a'}"]
\\
\as''
\ar[r, "1"]
\as''
\end{tikzcd}
\label{eq:hypercube-composition-law-non-inv}
\end{equation}

Pairing with the 0-dimensional hypercube $\bs$ and restricting to the $\Spin^c$ structures in the statement gives the non-involutive version of Proposition <ref>. Note that we can stabilize (<ref>) to obtain the following hypercube of attaching curves on $\Sigma\# \bar \Sigma$:

\begin{equation}
\begin{tikzcd}[
row sep=1cm,
column sep=3cm,
fill opacity=.7,
text opacity=1
\as\bar \bs
\ar[r, "\Theta_{\a'\bar \b,\a\bar \b}"]
\ar[d, "\Theta_{\a''\bar \b, \a\bar \b}"]
\as'\bar\bs
\ar[d, "\Theta_{\a''\bar \b,\a'\bar \b}"]
\\
\as''\bar \bs
\ar[r, "1"]
\as''\bar \bs
\end{tikzcd}
\label{eq:hypercube-composition-law-non-inv-stabilized}
\end{equation}

Our hyperbox for Proposition <ref> will be of size $(1,1,6)$, which we organize into 6 hypercubes $\scC_1,\dots, \scC_6$, which we stack to obtain a hyperbox. The cubes $\scC_1$ and $\scC_2$ are shown in Figure <ref>.

\[
\begin{tikzcd}[
column sep={2cm,between origins},
row sep=1.2cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;}
\CF(\as,\bs)
\ar[dr, "f_{\a\to \a''}^{\b}"]
\ar[ddd, "F_1^{\bar \b, \bar \b}"]
\ar[rr, "f_{\a\to \a'}^{\b}"]
&[.7 cm]
\ar[rd, "f_{\a\to \a''}^{\b}"]
\ar[ddd,"F_1^{\bar \b, \bar \b}"]
&[.7 cm]
\\
&[.7 cm]
\CF(\as'',\bs)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "\id"]
&[.7 cm]
\CF (\as'', \bs)
\ar[ddd,"F_1^{\bar \b, \bar \b}"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed, " h_{\a\to \a'\to \a''}^{\b}" ]
\\
\\
|[alias=A]|\CF(\as\bar \bs, \bs \bar \bs)
\ar[dr, "f_{\a \bar \b\to \a'' \bar \b}^{\b \bar \b}"]
\ar[rr, "f_{\a \bar \b\to \a'\bar \b}^{\b \bar \b}"]
\ar[drrr,dashed, "h_{\a\bar \b \to \a'\bar \b\to \a''\bar \b}^{\b\bar\b}"]
&[.7 cm]\,&
|[alias=B]|\CF(\as' \bar \bs, \bs \bar \bs)
\ar[dr, "f_{\a'\bar \b\to\a'' \bar \b}^{\b \bar \b}"]
\\
& [.7 cm]
|[alias=D]|\CF(\as''\bar \bs, \bs \bar \bs)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"F_1^{\bar \b, \bar \b}"]
\ar[rr, "\id"]
&[.7 cm]
|[alias=C]|\CF(\as''\bar \bs, \bs \bar \bs)
\end{tikzcd}
\]

\[
\begin{tikzcd}[
column sep={2cm,between origins},
row sep=1.2cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;}
|[alias=A]|\CF(\as\bar \bs,\bs \bar \bs)
\ar[dr, "f_{\a\bar\b\to \a'' \bar \b}^{\b \bar \b}"]
\ar[ddd, "f_{\a \bar \b}^{\b \bar \b\to \Dt}"]
\ar[rr, "f_{\a\bar \b\to \a'\bar \b}^{\b\bar \b}"]
\ar[dddrr,dashed]
&[.7 cm]
&|[alias=B]|\CF(\as'\bar \bs,\bs\bar \bs)
\ar[rd, "f_{\a'\bar \b\to \a''\bar \b}^{\b\bar \b}"]
\ar[ddd,"f_{\a'\bar \b}^{\b \bar \b\to \Dt}"]
\ar[ddddr,dashed]
&[.7 cm]
\\
&[.7 cm]
|[alias=D]|\CF(\as''\bar \bs,\bs\bar \bs)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, "\id"]
&[.7 cm]
|[alias=C]|\CF (\as''\bar \bs, \bs\bar \bs)
\ar[ddd,"f_{\a''\bar \b}^{\b \bar \b\to \Dt}"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed, " h_{\a\bar \b\to\a'\bar \b\to \a''\bar \b}^{\b\bar \b}" ]
\\
\\
\CF(\as\bar \bs, \Ds)
\ar[dr, "f_{\a \bar \b\to \a''\bar \b}^{\Dt}"]
\ar[rr, "f_{\a \bar \b\to \a'\bar \b}^{\Dt}"]
\ar[drrr,dashed, "h_{\a\bar \b \to \a'\bar \b\to \a''\bar \b}^{\b\bar\b}"]
&[.7 cm]\,&
\CF(\as' \bar \bs,\Ds)
\ar[dr, "f_{\a'\bar \b\to\a'' \bar \b}^{\Dt}"]
\\
& [.7 cm]
\CF(\as''\bar \bs, \Ds)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"f_{\a \bar \b}^{\b\bar \b\to \Dt}"]
\ar[rr, "\id"]
\ar[from=uuuul,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuul,dashed]
&[.7 cm]
\CF(\as''\bar \bs, \Ds)
\end{tikzcd}
\]

The hypercubes $\scC_1$ and $\scC_2$. The gray faces are stacked.

The second hypercube in Figure <ref> also has a length 3 map which is not drawn. In the figure, the top hypercube is a hypercube of stabilization, and the bottom hypercube is obtained by pairing the hypercube in (<ref>) with the 1-dimensional hypercube

\[
\begin{tikzcd}[column sep=2cm, labels=description]
\bs \bar \bs\ar[r, "\Theta_{\b \bar \b, \Dt}"]& \Ds.
\end{tikzcd}
\]

We now move to the construction of the hypercube $\scC_3$ and $\scC_4$, starting by constructing some auxiliary hypercubes. In Figure <ref>, we construct the 3-dimensional analog of the hypercube of beta-attaching curves in (<ref>). We next consider the hypercube of chain complexes obtained by pairing the cube in (<ref>) with the 1-dimensional cube

\[
\begin{tikzcd}[column sep=2cm, labels=description]
\Ds\ar[r, "\Theta_{\Dt, \a''\bar \a''}"]& \as'' \bar \as''.
\end{tikzcd}
\]

column sep=1.3cm,between origins,
row sep=.8cm,
fill opacity=.7,
text opacity=1,
[.7 cm]
' '
[.7 cm]

[.7 cm]
” ”
[rr,line width=2mm,dash,color=white,opacity=.7]

[.7 cm]
' ”
[from=ulll,line width=2mm,dash,color=white,opacity=.7]

[.7 cm]

[.7 cm]
[from=uuu,line width=2mm,dash,color=white,opacity=.7]
[rr, "1"]
[from=uuuul,line width=2mm,dash,color=white,opacity=.7]

[.7 cm]

An auxiliary hypercube of $\beta$-attaching curves used in the construction of $\scC_4$.

We reorganize the resulting cube slightly, to obtain the the diagram shown in Figure <ref>. Although the diagram in the figure is not technically a hypercube, since the bottom face is subdivided, if we compress the bottom face, we obtain a hypercube of chain complexes, which we call $\scC_3$.

\[
\begin{tikzcd}[
column sep={1.8cm,between origins},
row sep={1.5cm,between origins},
fill opacity=.7,
text opacity=1,
\CF(\as\bar \bs,\Ds)
\ar[ddd, equal]
\ar[rrrr, "f_{\a\bar \b\to \a'\bar \b}^{\Dt}"]
\ar[ddrr, "f_{\a\bar \b\to \a'' \bar \b}^{\Dt}"]
\ar[dddddrrrrrr,dotted]
\CF(\as'\bar \bs, \Ds)
\ar[ddrr, "f_{\a'\bar \b \to \a''\bar \b}^{\Dt}"]
\ar[ddd,equal]
\ar[dddddrr,dashed, "h_{\a'\bar \b\to \a''\bar \b}^{\Dt\to \a''\bar \a''}" ]
\\
\\
&&\CF(\as''\bar \bs,\Ds)
\ar[rrrr,equal]
\CF(\as'' \bar \bs, \Ds)
\ar[from=uullllll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uullllll, sloped,"h_{\a\bar \b\to \a'\bar \b \to \a'' \bar \b}^{\Dt}",dashed]
\\
\CF(\as\bar \bs, \Ds)
\ar[dr, "f_{\a \bar \b}^{\Dt\to \a''\bar \a''}"]
\ar[rrrr, "f_{\a \bar \b\to \a'\bar \b}^{\Dt}"]
\ar[drrrrr,dashed, sloped, "h_{\a\bar \b \to \a'\bar \b}^{\Dt\to \a'' \bar \a''}", pos=.7]
\CF(\as'\bar \bs,\Ds)
\ar[dr, "f_{\a'\bar \b}^{\Dt\to \a''\bar\a''}"]
\\
\CF(\as\bar \bs, \as'' \bar \as'')
\ar[rr, equal]
\ar[dr, "f_{\a\bar \b\to \a''\bar \b}^{\a''\bar \a''}"]
\CF(\as\bar \bs, \as''\bar \as'')
\ar[rr, "f_{\a\bar \b\to \a'\bar \b}^{\a''\bar \a''}"]
\ar[dr, "f_{\a \bar \b\to \a''\bar \b}^{\a''\bar \a''}"]
\ar[drrr,dashed, "h_{\a \bar \b\to \a'\bar \b\to \a''\bar \b}^{\a''\bar \a''}"]
\CF(\as'\bar \bs, \as'' \bar \as'')
\ar[dr, "f_{\a'\bar \b\to \a''\bar \b}^{\a''\bar \a''}"]
\\
\,&&
\CF(\as'' \bar \bs, \as'' \bar \as'')
\ar[rr,equal]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"f_{\a''\bar \b}^{\Dt\to \a''\bar \a''}", pos=.6]
\ar[from=uuuuull,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuuull,"h_{\a\bar \b\to\a''\bar \b}^{\Dt\to \a''\bar \a''}",dashed]
\CF(\as'' \bar \bs, \as'' \bar \as'')
\ar[rr,equal]
\CF(\as'' \bar \bs, \as'' \bar \as'')
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"f_{\a''\bar \b}^{\Dt\to \a''\bar \a''}"]
\end{tikzcd}
\]

The diagram $\scC_3$.

Next, we pair the top face of Figure <ref> with the cube

\[
\begin{tikzcd}[labels=description, column sep=2cm]
\as\bar \bs \ar[r,"\Theta_{\a\bar \b, \a'\bar \b}"] &\as' \bar \bs,
\end{tikzcd}
\]

and modify the resulting hypercube to obtain the back-most portion of the diagram shown on the top of Figure <ref>. This diagram is not technically a hypercube, since some of the faces are subdivided. Nonetheless, we can form a hypercube $\scC_4$ as follows. We view the diagram as being obtained by stacking two box-like diagrams (one on the back, and one on the front). Some of the faces of these boxes are subdivided, so we compress these faces. This gives two cube-diagrams. The hypercube relations for these cubes are straightforward to verify. For the back-most cube, they follow from the hypercube relations for the original cube described above. For the front cube, the hypercube relations are straightforward to verify directly, since all of the triangle maps involve stabilizations of Heegaard triples for small isotopies of the attaching curves. Hence, Proposition <ref> may be used to destabilize the relevant counts, and then the small translate theorem for triangles <cit.>*Proposition 11.1 may be used to identify the destabilized maps with nearest point maps.

We now describe the construction of the hypercube $\scC_5$. We first pair the 0-dimensional hypercube $\as \bar \bs$ with the hypercube in Figure <ref>. Then we rearrange, and obtain the diagram shown in the middle of Figure <ref>.

There is a final hyperbox involving 3-handles, which we compress to obtain the hypercube $\scC_6$, shown on the bottom of Figure <ref>. Stacking and compressing the hypercubes $\scC_1,\dots, \scC_6$ proves the statement.

\[
\scC_4:
\begin{tikzcd}[
column sep={1.8cm,between origins},
row sep=.5cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Number in  {A,B}
{\coordinate (\Number) at (\Number.center);}
\filldraw[black] (A) node[rectangle,fill=white,fill opacity=.9, text opacity=1]{$\CF(\as \bar \bs, \as''\bar \as'') $};
\filldraw[black] (B) node[rectangle,fill=white,fill opacity=.9, text opacity=1]{$ \CF(\as''\bar \bs,\as''\bar \as'')$};
\CF(\as\bar \bs,\Ds)
\ar[ddd, equal]
\ar[rrrr, ]
\ar[dr]
\ar[ddddrrrrr,dotted]
\ar[dddrrrr,dashed]
\CF(\as'\bar \bs, \Ds)
\ar[dr,]
\ar[ddd,]
\ar[ddddr,dashed,  ]
\\
&|[alias=A]| \phantom{\CF(\as \bar \bs, \as''\bar \as'')}
\ar[dr]
\ar[rr,equal]
\CF(\as \bar \bs, \as'' \bar \as'')
\ar[rr]
\ar[ddd]
\ar[dddrr,dashed]
\CF(\as'\bar \bs, \as'' \bar \as'')
\ar[dr]
\ar[ddd]
\ar[from=ulllll,dashed]
\\
\phantom{\CF(\as''\bar \bs,\as''\bar \as'')}
\ar[rrrr,equal,crossing over]
\CF(\as'' \bar \bs, \as''\bar \as'')
\ar[from=ulllll,dashed,crossing over]
\\
\CF(\as\bar \bs, \Ds)
\ar[dr]
\ar[rr]
\ar[drrr,dashed]
\CF(\as \bar \bs, \as'\bar \as')
\ar[rr]
\ar[dr]
\ar[drrr,dashed]
\CF(\as'\bar \bs,\as'\bar \as')
\ar[dr]
\\
\CF(\as\bar \bs, \as'' \bar \as'')
\ar[rr]
\ar[dr]
\ar[from=uuu, crossing over, equal]
\CF(\as\bar \bs, \as'\bar \as'')
\ar[rr]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=1,shorten <= 1.5cm]
\ar[from=uuu,shorten <= 1.5cm]
\CF(\as'\bar \bs, \as' \bar \as'')
\ar[dr, "F_3^{\a',\a'}"]
\\
\,&&
\CF(\as'' \bar \bs, \as'' \bar \as'')
\ar[rrrr,"F_3^{\a'',\a''}"]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,equal]
\CF(\bar \bs, \bar \as'')
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"F_3^{\a'',\a''}"]
\end{tikzcd}
\]

\[
\scC_5:
\begin{tikzcd}[
column sep={1.8cm,between origins},
row sep=.5cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Number in  {A,B}
{\coordinate (\Number) at (\Number.center);}
\filldraw[black] (A) node[rectangle,fill=white,fill opacity=.9, text opacity=1]{$\CF(\as \bar \bs, \as''\bar \as'') $};
\filldraw[black] (B) node[rectangle,fill=white,fill opacity=.9, text opacity=1]{$\CF(\as''\bar \bs,\as''\bar \as'')$};
\CF(\as\bar \bs,\Ds)
\ar[ddd]
\ar[rr ]
\ar[dr]
\ar[dddrr,dashed]
\ar[ddddrrr,dotted]
\CF(\as \bar \bs, \as'\bar \as')
\ar[dr]
\ar[rr]
\ar[ddd,shift left]
\ar[ddddr,dashed]
\CF(\as'\bar \bs, \as'\bar \as')
\ar[dr,]
\ar[ddd,"F_3^{\a',\a'}"]
\\
&|[alias=A]|\phantom{\CF(\as \bar \bs, \as''\bar \as'')}
\ar[dr]
\ar[rr,crossing over]
\ar[ddd]
\CF(\as \bar \bs, \as' \bar \as'')
\ar[rr,crossing over]
\ar[ddd]
\ar[from=ulll,dashed,crossing over]
\CF(\as'\bar \bs, \as' \bar \as'')
\ar[dr, "F_3^{\a',\a'}"]
\ar[ddd, "F_3^{\a',\a'}"]
\ar[from=ulll,dashed,crossing over]
\\
|[alias=B]|\phantom{\CF(\as''\bar \bs,\as''\bar \as'')}
\ar[rrrr,crossing over,"F_3^{\a'',\a''}", pos=.3]
\CF(\bar \bs, \bar \as'')
\\
\CF(\as\bar \bs, \as\bar \as)
\ar[dr]
\ar[rr]
\ar[drrr,dashed]
\CF(\as \bar \bs, \as\bar \as')
\ar[rr, "F_3^{\a,\a}", pos=.7]
\ar[dr]
\CF(\bar \bs,\bar \as')
\ar[dr]
\\
\CF(\as\bar \bs, \as \bar \as'')
\ar[rr,equal]
\ar[dr, "F_3^{\a,\a}"]
\ar[from=uuu, crossing over]
\ar[from=uuuul,crossing over,dashed]
\CF(\as\bar \bs, \as\bar \as'')
\ar[rr, "F_3^{\a,\a}"]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=1, shorten <= 2cm]
\ar[from=uuu, shorten <= 2cm]
\CF(\bar \bs,\bar \as'')
\ar[dr, equal]
\\
\,&&
\CF(\bar \bs,\bar \as'')
\ar[rrrr,equal]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"F_3^{\a'',\a''}", pos=.85]
\CF(\bar \bs, \bar \as'')
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,equal]
\end{tikzcd}
\]

\[
\scC_6:
\begin{tikzcd}[
column sep={1.8cm,between origins},
row sep=.5cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Number in  {A}
{\coordinate (\Number) at (\Number.center);}
\filldraw[black] (A) node[rectangle,fill=white,fill opacity=.9, text opacity=1]{$\CF(\bar \bs,\bar \as'')$};
\CF(\as\bar \bs,\as \bar \as)
\ar[ddd, "F_3^{\a, \a}"]
\ar[rr]
\ar[dr]
\CF(\as \bar \bs, \as\bar \as')
\ar[dr]
\ar[rr, "F_3^{\a,\a}"]
\ar[ddd, "F_3^{\a,\a}",shift left]
\CF(\bar \bs, \bar \as')
\ar[dr,]
\ar[ddd,equal]
\\
& \CF(\as \bar \bs, \as\bar \as'')
\ar[dr, "F_3^{\a,\a}"]
\ar[rr,crossing over, equal]
\CF(\as \bar \bs, \as\bar \as'')
\ar[rr, "F_3^{\a,\a}"]
\ar[ddd, "F_3^{\a,\a}"]
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed]
\CF(\bar \bs, \bar \as'')
\ar[dr,equal]
\ar[ddd,equal]
\\
|[alias=A]|\phantom{\CF(\bar \bs,\bar \as'')}
\ar[rrrr,crossing over,equal]
\CF(\bar \bs, \bar \as'')
\\
\CF(\bar \bs,\bar \as)
\ar[dr]
\ar[rr]
\ar[drrr,dashed]
\CF(\bar \bs, \bar \as')
\ar[rr, equal]
\ar[dr]
\CF(\bar \bs,\bar \as')
\ar[dr]
\\
\CF(\bar \bs, \bar \as'')
\ar[rr,equal]
\ar[dr,equal]
\ar[from=uuu,crossing over, "F_3^{\a,\a}"]
\CF(\bar \bs,\bar \as'')
\ar[rr, equal]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=1, shorten <= 1.7cm]
\ar[from=uuu,shorten <= 1.7cm]
\CF(\bar \bs,\bar \as'')
\ar[dr, equal]
\\
\,&&
\CF(\bar \bs,\bar \as'')
\ar[rrrr,equal]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7,shift right=.3mm]
\ar[from=uuu,equal, shift right=.3mm]
\CF(\bar \bs, \bar \as'')
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,equal]
\end{tikzcd}
\]

The diagrams used to construct $\scC_4$, $\scC_5$ and $\scC_6$

§.§ More on commutations of handles

In this section, we prove that 1-handle and 3-handle maps may be commuted with 2-handle maps, when the corresponding topological handle manipulation is also possible:

Suppose that $\bL$ is a framed 1-dimensional link in $Y$, and $\bS$ is a 0- or 2-dimensional sphere in $Y\setminus \bL$. Let $\frS\subset \Spin^c (W(Y,\bL))$ be a set of $\Spin^c$ structures which is closed under conjugation, and let $\frS'\subset \Spin^c (W(Y(\bS),\bL))$ be the corresponding set. Then

\[
\CFI(W(Y(\bL),\bS))\circ \CFI(W(Y,\bL),\frS)\simeq \CFI(W(Y(\bS),\bL),\frS')\circ \CFI(W(Y,\bS)).
\]

The proof is similar to the proof of Proposition <ref>. We will build a 3-dimensional hypercube, four of whose faces correspond to the four maps $\CFI(W(Y(\bL),\bS))$,  $\CFI(W(Y,\bL),\frS)$,  $\CFI(W(Y(\bS),\bL),\frS')$, and $\CFI(W(Y,\bS))$. The hypercube relations for the compression of this hyperbox translate exactly to the chain homotopy for the commutation described in the statement.

The hypercube realizing the commutation will be constructed as the compression of a hyperbox of size $(1,1,5)$, which we view as five hypercubes, stacked on top of each other. We write $\scB_1,\cdots \scB_5$ for these hypercubes.

The top-most hypercubes, $\scB_1$ and $\scB_2$, are shown in Figure <ref>. They are constructed by stacking hypercubes constructed by pairing hypercubes of attaching curves, as well as hypercubes of stabilization.

\[
\scB_1:\hspace{-.5cm}
\begin{tikzcd}[
column sep={2.1cm,between origins},
row sep=1cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;}
\CF(\as,\bs)
\ar[dr,"f_{\a\to \a'}^{\b}"]
\ar[ddd, "F_1^{\bar \b,\bar \b}"]
\ar[rr, "F_1^{\g_0,\g_0}"]
&[.7 cm]
\ar[rd, "f_{\a\g_0\to \a'\g_0}^{\b\g_0}"]
\ar[rr,equal]
\ar[ddd, "F_1^{\bar\b\bar \g_0,\bar\b\bar \g_0}"]
&[.7 cm]
\CF(\as\g_0,\bs \g_0)
\ar[dr, "f_{\a\g_0\to \a'\g_0}^{\b\g_0}"]
\ar[ddd,"F_1^{\bar \b \bar \g_0, \bar \b \bar \g_0}"]
&[.7 cm]
\\
&[.7 cm]
\CF(\as',\bs)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr," F_1^{\g_0,\g_0}"]
&[.7 cm]
\CF(\as'\g_0,\bs\g_0)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, equal]
\CF (\as'\g_0,\bs\g_0)
\ar[ddd,"F_1^{\bar \b \bar \g_0, \bar \b \bar \g_0}"]
\\
\\
|[alias=A]| \CF(\as \bar \bs, \bs \bar\bs)
\ar[dr, "f_{\a\bar \b\to \a'\bar \b}^{\b\bar \b}"]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
&[.7 cm]\,&
\CF(\as \bar \bs \g_0\bar \g_0, \bs \bar \bs \g_0 \bar \g_0)
\ar[dr, "f_{\a\bar \b\g_0\bar \g_0\to \a'\bar \b \g_0\bar \g_0}^{\b \bar \b \g_0\bar \g_0}"]
\ar[rr,equal]
&[.7 cm]\,
|[alias=B]|  \CF(\as \bar \bs \g_0 \bar \g_0, \bs \bar \bs \g_0 \bar \g_0)
\ar[dr,"f_{\a\bar \b\g_0\bar \g_0\to \a'\bar \b \g_0\bar \g_0}^{\b \bar \b \g_0\bar \g_0}"]
\\
& [.7 cm]
|[alias=D]|  \CF(\as'\bar \bs, \bs \bar \bs)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,"F_1^{\bar \b,\bar \b}"]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
&[.7 cm]
\CF(\as'\bar \bs \g_0\bar \g_0, \bs \bar \bs \g_0\bar \g_0)
\ar[rr,equal]
\ar[from=uuu, crossing over, "F_1^{\bar \b \bar \g_0, \bar \b \bar \g_0}"]
|[alias=C]| \CF(\as'\bar \bs\g_0\bar \g_0,\bs \bar \bs \g_0\bar \g_0)
\end{tikzcd}
\]

\[
\scB_2:\hspace{-.5cm}
\begin{tikzcd}[
column sep={2.1cm,between origins},
row sep=1cm,
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;}
|[alias=A]| \CF(\as\bar \bs,\bs\bar \bs)
\ar[dr]
\ar[ddd]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
&[.7 cm]
&\CF(\as\bar\bs \g_0\bar \g_0,\bs\bar \bs\g_0\bar \g_0)
\ar[rd]
\ar[ddd]
\ar[rr,equal]
\ar[ddddr,dashed]
&[.7 cm]
|[alias=B]|  \CF(\as\bar \bs\g_0\bar \g_0,\bs\bar \bs \g_0\bar \g_0)
\ar[dr]
\ar[ddd]
\ar[ddddr,dashed]
&[.7 cm]
\\
&[.7 cm]
|[alias=D]|\CF(\as'\bar \bs,\bs \bar \bs)
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr," F_1^{\g_0 \bar \g_0,\g_0 \bar \g_0}"]
&[.7 cm]
\CF(\as'\bar \bs\g_0\bar \g_0,\bs\bar \bs\g_0\bar \g_0)
\ar[ddd]
\ar[rr,line width=2mm,dash,color=white,opacity=.7]
\ar[rr, equal]
|[alias=C]|   \CF(\as'\bar \bs\g_0\bar \g_0,\bs\bar \bs\g_0\bar \g_0)
\ar[ddd]
\\
\\
\CF(\as \bar \bs,\Ds)
\ar[dr,]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
&[.7 cm]\,&
\CF(\as \bar \bs \g_0\bar \g_0, \Ds \g_0 \bar \g_0)
\ar[dr,]
\ar[rr,]
\ar[drrr,dashed]
&[.7 cm]\,
\CF(\as \bar \bs \g_0 \bar \g_0, \Ds \Ds_0)
\ar[dr]
\\
& [.7 cm]
\CF(\as'\bar \bs, \Ds)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,]
\ar[rr,"F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[from=uuuul,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuul,dashed]
&[.7 cm]
\CF(\as'\bar \bs \g_0\bar \g_0, \Ds\g_0\bar \g_0)
\ar[rr]
\ar[from=uuu,crossing over]
\CF(\as'\bar \bs\g_0\bar \g_0,\Ds \Ds_0)
\end{tikzcd}
\]

The hypercubes $\scB_1$ (top) and $\scB_2$ (bottom) are obtained by compressing these two diagrams. The unlabeled arrows are all holomorphic triangle or quadrilateral maps.    There is also a length 3 arrow in the right subcube for $\scB_2$ which is not drawn.

The diagram used to construct $\scB_3$ is shown in Figure <ref>.

\[
\begin{tikzcd}[
column sep={2.2cm,between origins},
row sep={1.5cm,between origins},
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Number in  {A,B}
{\coordinate (\Number) at (\Number.center);}
\filldraw[black] (A) node[rectangle,fill=white, fill opacity=.9, text opacity=1]{$\CF(\as' \bar \bs, \Ds)$};
\filldraw[black] (B) node[rectangle,fill=white,fill opacity=.9, text opacity=1]{$\CF(\as'\bar \bs \g_0\bar \g_0, \Ds \g_0\bar \g_0)$};
\CF(\as\bar \bs,\Ds)
\ar[ddd,equal]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
\ar[ddrr]
\CF(\as \bar \bs \g_0 \bar \g_0, \Ds\g_0\bar \g_0)
\ar[rr]
\ar[ddd,dash, shift left=2mm]
\ar[ddd,dash, shift left=1.3mm]
\ar[ddrr]
\ar[dddddrr,dashed]
\CF(\as\bar \bs \g_0 \bar \g_0, \Ds \Ds_0)
\ar[ddrr]
\ar[ddd,shift left=2mm,dash]
\ar[ddd,shift left=1.3mm, dash]
\ar[dddddrr,dashed]
\\
\\
&&|[alias=A]|\phantom{\CF(\as' \bar \bs, \Ds)}
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
|[alias=B]|\phantom{\CF(\as'\bar \bs \g_0\bar \g_0, \Ds \g_0\bar \g_0)}
\ar[rr]
\CF(\as'\bar \bs \g_0 \bar \g_0, \Ds\Ds_0)
\ar[from=uullll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uullll,dashed]
\\
\CF(\as\bar \bs, \Ds)
\ar[dr]
\ar[rr,"F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\CF(\as\bar \bs \g_0 \bar \g_0, \Ds \g_0\bar \g_0)
\ar[rr]
\ar[dr]
\ar[drrr,dashed]
\CF(\as \bar \bs \g_0 \bar \g_0, \Ds \Ds_0)
\ar[dr]
\\
\CF(\as\bar \bs, \as' \bar \as')
\ar[rr, "F_1^{\g_0\bar\g_0,\g_0\bar\g_0}"]
\ar[dr]
\CF(\as\bar \bs \g_0 \bar \g_0, \as' \bar \as'\g_0\bar \g_0)
\ar[rr,"B_\tau"]
\ar[dr]
\ar[drrr,dashed]
\CF(\as \bar \bs \g_0 \bar \g_0, \as'\bar \as' \g_0 \bar \g_0)
\ar[dr]
\\
\CF(\as'\bar \bs, \as'\bar \as')
\ar[rr,"F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu]
\ar[from=uuuuull,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuuuull,dashed]
\CF(\as'\bar \bs \g_0 \bar \g_0, \as'\bar \as' \g_0\bar \g_0)
\ar[rr, "B_\tau"]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu]
\CF(\as'\bar \bs \g_0 \bar \g_0, \as'\bar \as' \g_0 \bar \g_0)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu]
\ar[from=uuull,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuull,dashed]
\end{tikzcd}
\]

The diagram used to construct the hypercube $\scB_3$.

The hypercube $\scB_4$ is displayed in Figure <ref>. The construction is as follows. The back-left cube is a hypercube of stabilization for the map $F_1^{\g_0\bar \g_0, \g_0\bar \g_0}$. The back-right cube is constructed by pairing the 0-dimensional hypercube $\as \bar \bs \g_0\bar \g_0$ with a 3-dimensional generalized hypercube of beta attaching curves which has $\tau$ as one of the length 1 morphisms. The construction is similar to the definition of the 1-handle map, and the proof of well definedness is similar to Lemma <ref>. The front portion is verified to satisfy the hypercube relations (after compressing the top face) by observing that the length 2 map composes trivially with the 3-handle map $F_3^{\a'\g_0,\a'\g_0}$ by applying our stabilization  result from Proposition <ref> and applying the small translate theorems for triangles and quadrilaterals <cit.>.

Our final hypercube $\scB_5$ is shown in Figure <ref>. Stacking and compressing these hypercubes completes the proof.

\[
\begin{tikzcd}[
column sep={2.3cm,between origins},
row sep={1.8cm,between origins},
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;}
\CF(\as\bar \bs,\Ds)
\ar[dd]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
\ar[dr]
\CF(\as \bar \bs \g_0 \bar \g_0, \Ds\g_0\bar \g_0)
\ar[rr]
\ar[dd, shift left=1mm]
\ar[dr]
\ar[dddr,dashed]
\ar[ddrr,dashed]
\CF(\as\bar \bs \g_0 \bar \g_0, \Ds \Ds_0)
\ar[dr]
\ar[dd,shift left=1mm]
\ar[dddr,dashed]
\\
|[alias=A]|\CF(\as \bar \bs, \as'\bar \as')
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
\CF(\as \bar \bs \g_0\bar \g_0, \as'\bar \as' \g_0\bar \g_0)
\ar[rr, "B_\tau"]
\ar[ddrr,dashed]
|[alias=B]|\CF(\as \bar \bs \g_0\bar \g_0, \as'\bar \as' \g_0\bar \g_0)
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed]
\ar[dd]
\\[.5cm]
\CF(\as\bar \bs, \as \bar \as)
\ar[dr]
\ar[rr,"F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\CF(\as \bar \bs \g_0 \bar \g_0, \as \bar \as \g_0 \bar \g_0)
\ar[rr, "B_\tau"]
\ar[dr]
\ar[drrr,dashed]
\CF(\as \bar \bs \g_0 \bar \g_0,\as \bar \as \g_0 \bar \g_0)
\ar[dr]
\\
|[alias=D]|\CF(\as\bar \bs, \as \bar \as')
\ar[rr, "F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[from=uu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uu]
\ar[from=uuul,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuul,dashed]
\CF(\as\bar \bs \g_0 \bar \g_0, \as \bar \as'\g_0\bar \g_0)
\ar[rr,"B_\tau"]
\ar[from=uu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uu]
|[alias=C]|\CF(\as \bar \bs \g_0 \bar \g_0, \as\bar \as' \g_0 \bar \g_0)
\end{tikzcd}
\]

\[
\begin{tikzcd}[
column sep={2.3cm,between origins},
row sep={1.8cm,between origins},
fill opacity=.7,
text opacity=1,
execute at end picture={
\foreach \Nombre in  {A,B,C,D}
{\coordinate (\Nombre) at (\Nombre.center);}
\fill[opacity=0.1]
(A) -- (B) -- (C) -- (D) -- cycle;},
|[alias=A]|\CF(\as \bar \bs, \as'\bar \as')
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
\CF(\as \bar \bs \g_0\bar \g_0, \as'\bar \as' \g_0\bar \g_0)
\ar[rr, "B_\tau"]
\ar[ddrr,dashed]
\ar[dd]
|[alias=B]|\CF(\as \bar \bs \g_0\bar \g_0, \as'\bar \as' \g_0\bar \g_0)
\ar[dd]
\\
&\CF(\as' \bar \bs, \as'\bar \as')
\ar[rr,crossing over, "F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[from=ul]
\CF(\as'\bar \bs \g_0\bar \g_0, \as'\bar \as' \g_0\bar \g_0)
\ar[rr,crossing over,pos=.3, "B_\tau"]
\ar[from=ul]
\CF(\as'\bar \bs \g_0 \bar \g_0, \as'\bar \as' \g_0 \bar \g_0)
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed]
\ar[from=ul]
\\[.5cm]
|[alias=D]|\CF(\as\bar \bs, \as \bar \as')
\ar[rr, "F_1^{\g_0\bar \g_0, \g_0\bar \g_0}"]
\ar[dr, "F_3^{\a,\a}"]
\ar[from=uu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uu]
\CF(\as\bar \bs \g_0 \bar \g_0, \as \bar \as'\g_0\bar \g_0)
\ar[rr,"B_\tau"]
|[alias=C]|\CF(\as \bar \bs \g_0 \bar \g_0, \as\bar \as' \g_0 \bar \g_0)
\ar[dr, "F_3^{\a \g_0,\a \g_0}"]
\\
\,&
\CF(\bar \bs, \bar \as')
\ar[rrrr,"F_1^{\bar \g_0,\bar \g_0}"]
\ar[from=uu, "F_3^{\a',\a'}",crossing over, pos=.3]
\CF(\bar \bs \bar \g_0, \bar \as' \bar \g_0)
\ar[from=uu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uu, "F_3^{\a'\g_0,\a'\g_0}"]
\end{tikzcd}
\]

The diagram used to build the hypercube $\scB_4$. We stack the diagrams along the gray face. There is a length 3 map in the right portion of top-most diagram which is not drawn.

\[
\begin{tikzcd}[
column sep={2.2cm,between origins},
row sep={1.1cm,between origins},
fill opacity=.7,
text opacity=1,
\CF(\as \bar \bs, \as \bar \as)
\ar[ddd, "F_3^{\a,\a}"]
\ar[rr, "F_1^{\g_0\bar \g_0,\g_0\bar \g_0}"]
\ar[dr]
\CF(\as \bar \bs \g_0 \bar \g_0, \as \bar \as\g_0\bar \g_0)
\ar[rr, "B_\tau"]
\ar[dr]
\CF(\as\bar \bs \g_0 \bar \g_0, \as \bar \as \g_0 \bar \g_0)
\ar[dr]
\ar[ddd,shift left=1mm]
\\
\CF(\as \bar \bs, \as \bar \as')
\ar[rr, "F_1^{\g_0\bar\g_0,\g_0\bar \g_0}"]
\CF(\as \bar \bs \g_0\bar \g_0, \as\bar \as' \g_0\bar \g_0)
\ar[rr, "B_\tau",crossing over, pos=.3]
\CF(\as \bar \bs \g_0\bar \g_0, \as\bar \as' \g_0\bar \g_0)
\ar[from=ulll,line width=2mm,dash,color=white,opacity=.7]
\ar[from=ulll,dashed]
\\
&&\CF(\bar \bs, \bar \as')
\ar[rrrr, "F_1^{\bar \g_0, \bar \g_0}", pos=.45, crossing over]
\ar[from=ul, "F_3^{\a,\a}"]
\CF(\bar \bs \bar \g_0, \bar \as' \bar \g_0)
\ar[from=ul, "F_3^{\a,\a}"]
\\
\CF(\bar \bs, \bar \as)
\ar[ddrr]
\ar[rrrr,"F_1^{\bar \g_0,\bar \g_0}", pos=.4]
\CF(\bar \bs \bar \g_0, \bar \as \bar \g_0)
\ar[ddrr]
\\
\,
\\
\,
\,&
\CF(\bar \bs, \bar \as')
\ar[rrrr,"F_1^{\bar \g_0,\bar \g_0}"]
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu, equal]
\CF(\bar \bs \bar \g_0, \bar \as' \bar \g_0)
\ar[from=uuu,line width=2mm,dash,color=white,opacity=.7]
\ar[from=uuu,equal]
\end{tikzcd}
\]

The diagram used to construct the hypercube $\scB_5$

§ NATURALITY MAPS FOR STABILIZATIONS

In this section, we construct our naturality maps for stabilizations of the Heegaard surface, and prove some basic properties. Our definition of the naturality map for a stabilization is simple: we define the map for a stabilization as the composition of a 1-handle cobordism map with the 2-handle cobordism map for a 2-handle which topologically cancels the 1-handle.

The definition gives some diagrammatic flexibility for computing the stabilization map. In particular, for a stabilization we require two points $p_0$ and $p_1$ to be chosen on $\Sigma\setminus (\as \cup \bs)$. We then add a tube to $\Sigma$ with feet at $p_0$ and $p_1$. We add one new alpha curve and one beta curve. The beta curve is a meridian of the tube, and the alpha curve is obtained by concatenating a longitude of the curve with an embedded curve on $\Sigma\setminus \as$. If $\Sigma$ is embedded in $Y$, then we may view the construction as being determined by picking an embedded path on $\Sigma\setminus \as$ which connects $p_0$ and $p_1$, and having the tube run parallel to this path. The new alpha circle runs parallel to this arc. We define the destabilization map as the composition of a 2-handle map for a 0-framed unknot, followed by the 3-handle cobordism map.

Note that this stabilization operation naturally determines a 3-ball in $Y$ which intersects $\Sigma$ in a disk. We say two stabilizations are disjoint if there are two choices of corresponding 3-balls in $Y$ which are disjoint.

We write $\sigma$ for the stabilization map, and $\tau$ for the destabilization map. The main results of this section are summarized in the following proposition.

* The stabilization and destabilization maps commute with the transition maps for elementary equivalences of the doubling-enhanced Heegaard diagram.

*  If $\sigma$ and $\sigma'$ are the maps for two disjoint stabilizations, then

\[
\sigma\circ \sigma'\simeq \sigma'\circ \sigma.
\]

*  If $\sigma$ and $\tau$ are stabilizations and destabilizations for the same 3-ball, then

\[
\sigma\circ \tau\simeq \id\quad \text{and} \quad \tau\circ \sigma\simeq \id.
\]

Our proof of Parts (<ref>) and (<ref>) goes by way of computing basepoint-adjacent stabilizations and destabilizations (note that by handlesliding the original alpha and beta curves, such a configuration may always be achieved):

Suppose that $p_1$ and $p_0$ are two points on $\Sigma\setminus (\as\cup \bs)$ which both lie in the same component as the basepoint $w$. For a suitable choice of doubling curves on the stabilized diagram and suitably degenerated choices of almost complex structures, we have

\[
\sigma\simeq (\xs\mapsto \xs\times c)\quad \text{and} \quad \tau\simeq(\xs\times c\mapsto \xs),
\]

where $\{c\}=\a_0\cap \b_0$, and $(\xs\mapsto \xs\times c)$ denotes the $\bF[U,Q]/Q^2$-equivariant map which sends $\xs$ to $\xs\times c$, and similarly for $(\xs\times c\mapsto \xs)$.

Consider $\sigma$ (the proof for $\tau$ is essentially the same). By definition, $\sigma$ is the composition of the 1-handle cobordism map, followed by the map for a canceling 2-handle. By Lemma <ref>, for a suitable choice of almost complex structure, the 1-handle map takes the form $\xs\mapsto \xs\times \theta^+$, extended $\bF[U,Q]/Q^2$-equivariantly. We will show that the 2-handle cobordism map is chain homotopic to a map which sends $\xs\times \theta^+$ to $\xs\times c$.

We recall that the 2-handle map is defined in Equation (<ref>). We claim that the diagonal map of the compression is trivial. To see this, we make the following subclaims about the diagram in (<ref>) in the present context:

*  The top-most length 2 map vanishes on elements of the form $\xs\times \theta_{\b_0,\b_0}^+\times \Theta_{\bar \b, \bar \b}^+\times \theta_{\bar \b_0,\bar \b_0}^+$, for $\xs\in \bT_{\a}\cap \bT_{\b}$.

*  The middle length 2 map composes trivially with the 3-handle map $F_3^{\a  \b_0, \a  \b_0}$ (labeled $F_3^{\a',\a'}$ in Equation (<ref>)), and the bottom-most length 2 map composes trivially with the map $F_3^{\a \a_0, \a \a_0}$ (labeled $F_3^{\a,\a}$ in Equation (<ref>)).

We consider claim (<ref>) first. There is a genus 2 portion of the Heegaard quadruple corresponding to the stabilization, which takes the following form:

\[
\cQ_0=(\bT^2\# \bT^2, \a_0 \bar \b_0, \b_0 \bar \b_0, \b_0 \bar \b_0, \Ds_0).
\]

We observe that

\[
\widehat{f}_{\a_0\bar \b_0, \b_0\bar \b_0, \Dt_0}(\widehat{f}_{\a_0\bar \b_0,\b_0\bar \b_0, \b_0\bar \b_0}(\Theta_{\a_0\bar \b_0, \b_0\bar \b_0}^+, \Theta_{\b_0\bar \b_0, \b_0\bar \b_0}^+), \Theta_{\b_0\bar \b_0, \Dt_0}^+)=\Theta_{\a_0\bar \b_0, \Dt_0}.
\]

This may be seen since the inner triangle map corresponds to the naturality map for a small perturbation of the curves $\b_0\bar \b_0$, while the outer triangle map corresponds to a factor of the doubling model of the involution for the diagram $(\bT^2, \a_0,\b_0)$. Hence, by applying Proposition <ref>, we obtain the equality

\[
h_{\a \b_0 \bar \b \bar \b_0\to \a \a_0 \bar \b \bar \b_0}^{\b \b_0 \bar \b \bar \b_0\to \Dt\Dt_0}(\xs\times \theta_{\b_0,\b_0}^+\times \Theta_{\bar \b, \bar \b}^+\times \theta_{\bar \b_0,\bar \b_0}^+)=h_{\a \bar \b \to \a \bar \b}^{\b \bar \b \to \Dt}(\xs\times \theta_{\b_0,\b_0}^+)\otimes \Theta_{\a_0\bar \b_0, \Dt_0}.
\]

which vanishes by the small translate theorem for holomorphic quadrilaterals <cit.>.

Next, we consider claim (<ref>), from above. There are two quadrilateral maps which appear in this claim. The proof for both maps is essentially the same, so we focus on the map

\[
h_{\a \b_0\bar \b \bar \b_0\to \a \a_0 \bar \b\bar \b_0}^{\Dt\Dt_0\to \a  \a_0\bar \a \bar \a_0}.
\]

This is the middle length 2 map in (<ref>). We wish to show that this map composes trivially with $F_3^{\a',\a'}$. To see this, we consider the genus 2 Heegaard quadruple

\[
\cQ_1= (\bT^2\# \bT^2, \a_0 \bar \a_0, \Dt_0, \b_0\bar \b_0, \a_0\bar \b_0).
\]

This is an admissible, algebraically rigid multi-stabilizing quadruple. Furthermore, we claim that

\begin{equation}
\widehat{f}_{\a_0\bar \a_0, \b_0\bar \b_0, \a_0\bar \b_0}(\widehat{f}_{\a_0\bar \a_0,\Dt_0, \b_0\bar \b_0}(\Theta^+_{\a_0\bar \a_0, \b_0\bar \b_0},\Theta^+_{\Dt_0, \b_0\bar \b_0}),\Theta^+_{\b_0\bar \b_0, \a_0\bar \b_0})=\Theta^-_{\a_0\bar \a_0, \a_0\bar \b_0}.
\label{eq:model-computation-stabilization-2-handle}
\end{equation}

The above equation is verified by noting that the inner triangle map may be interpreted as the cobordism map for a 2-handle attachment along a fiber of $S^1\times S^2$, while the outer triangle map may be interpreted as a 2-handle attached along a 0-framed unknot. The fact that $\Theta^-_{\a_0\bar \a_0, \a_0\bar \b_0}$ is the output may be viewed as a consequence of the above topological description.

Combining (<ref>) with Proposition <ref>, we conclude that

\begin{equation}
h_{\a \b_0\bar \b \bar \b_0\to \a \a_0 \bar \b\bar \b_0}^{\Dt\Dt_0\to \a  \a_0\bar \a \bar \a_0}(\xs \times \Theta^+_{\b_0\bar \b_0,\Dt})= h_{\a \bar \b \to \a \bar \b}^{\Dt \to \a \bar \a}(\xs)\times \Theta^-_{\a_0\bar \b_0, \a_0 \bar \a_0}+\sum_{\zs\in \bT_{\a\bar \b}\cap \bT_{\a \bar \a}} C_{\zs}\cdot \zs\times \Theta_{\a_0\bar \b_0,\a_0\bar \a_0}^+,
\label{eq:destabilize-quadrilatal-model-comp-stabilization}
\end{equation}

for some $C_{\zs}\in \bF[U]$. Equation  (<ref>) vanishes once composed with $F_{3}^{\a \a_0,\a \a_0}$, by the small translate theorem for holomorphic quadrilaterals <cit.>.

The composition of the bottom-most length 2 map from Equation (<ref>) with $F_3^{\a \b_0,\a\b_0}$ vanishes in our present setting by a nearly identical argument.

We can now prove Proposition <ref>:

Claim (<ref>) follows immediately from the fact that 1-handle, 2-handle and 3-handle cobordism maps commute (up to chain homotopy) with the transition maps for elementary handleslides.

Claims (<ref>) and (<ref>) follow from Lemma <ref>, since by (<ref>), we may perform stabilizations near the basepoint, and then observe that the formulas given therein clearly satisfy the stated relations.

§ HANDLESWAPS

In this section, we prove handleswap invariance.

§.§ Simple handleswaps

Of central importance to the approach to naturality in <cit.>  are the following loops in the graph of Heegaard diagrams for a 3-manifold $Y$:

Suppose $Y$ is a 3-manifold. A simple handleswap loop  is a triangle of Heegaard diagrams

\[
\begin{tikzcd}
&\cH_1\arrow[rd,"e"] & \\
\cH_3 \arrow[ur,"g"] && \cH_2 \arrow[ll,"f"]
\end{tikzcd}
\]

which satisfies the following:

* $\cH_i $ is an embedded Heegaard diagram for $Y$, which has the same Heegaard surface for each $i\in \{1,2,3\}$.

* $e$ is an $\a$-equivalence, $f$ is a $\b$-equivalence, and $g$
is a diffeomorphism.
\item	The diagrams $_i$ decompose as connected sums of a fixed Heegaard diagram $=(Σ,_u,_u)$ with genus two diagrams $_i=(Σ_0,_i,_i)$. Furthermore, in the punctured genus two surface $(Σ# Σ_0)∖Σ$, the above triangle is diffeomorphic to the  triangle in Figure~\ref{fig:handlewap-elem}.
\end{enumerate}
\end{define}

In particular, in the above definition, $_1$ consists of two closed curves $_1$ and $_2$, while $_1$ consists of two closed curves $_̱1$ and $_̱2$. The arrow $e$ corresponds to handlesliding $_1$ over $_2$, while the arrow $f$ corresponds to handlesliding $_̱1$ over $_̱2$.

\begin{figure}[H]
\centering
\input{handleswap-elem.pdf_tex}
\caption{A simple handleswap.}\label{fig:handlewap-elem}
\end{figure}

Recall that associated to handleslides and diffeomorphisms, we have defined morphisms, $(e)$, $(f)$ and $(g)$, which are well-defined up to $[U,Q]/Q^2$-equivariant homotopy equivalence.

\begin{thm}
\label{thm:handleswap}
\CFI(g)\circ \CFI(f)\circ \CFI(e)\simeq \Id_{\CFI(\cH_1)}.$

\end{thm}

The proof of Theorem~\ref{thm:handleswap} occupies the subsequent sections.

\subsection{Reduction to basepoint-adjacent handleswaps}

A simple handleswap need not occur in a region of the Heegaard diagram near a basepoint. However we now show that for the purposes of naturality, it is sufficient to consider handleswaps where the special genus 2 region is adjacent to the basepoint. We note that this fact was also observed by Sungkyung Kang.

\begin{lem}
Suppose that $F$ is a weak Heegaard invariant (in the sense of \cite{JTNaturality}*{Definition~2.24}) which satisfies all of the conditions of a strong Heegaard invariant (\cite{JTNaturality}*{Definition~3.23}) except possibly handleswap invariance. If $F$ has no monodromy around basepoint-adjacent handleswaps, then it has no monodromy around arbitrary handleswaps.
\end{lem}
\begin{proof}
Let $p\in \Sigma$ denote the connected sum point. Pick an arc $\lambda$ which connects $p$ to the basepoint $w$. We proceed by induction on the number of intersections of $(\as_u\cup \bs_u) \cap \lambda.$ The base case is the hypothesis. If $\cH$ is arbitrary, there is an equivalence $j$ between $\cH$ and a diagram $\cH'$ (obtained by handlesliding an alpha or beta curve over $p$) such that on $\cH'$, there is one fewer intersection of $(\as_u \cup \bs_u)\cap \lambda$. If $\cH'_1,$ $\cH_2'$ and $\cH_3'$ denote the diagrams obtained by connect summing the standard genus 2 diagrams from Figure~\ref{fig:handlewap-elem}, there is an analogous simple handleswap loop involving $\cH_1'$, $\cH_2'$ and $\cH_3'$. Write $e'$, $f'$ and $g'$ for the analogous equivalences. We have a diagram of equivalences
\[
\begin{tikzcd}[labels=description, column sep=.7cm, row sep=.9cm]
\ar[dddrrr, "e"]
\ar[d, "j_1"]
\ar[dr, "e'"]
\\
\ar[ur, "g'"]
\ar[from=dll, "j_3"]
&& \cH_2'
\ar[ll, "f'"]
\ar[drr, "j_2"]
\\[-.3cm]
\cH_3
\ar[uuurrr, "g"]
\cH_2
\ar[llllll, "f"]
\end{tikzcd}
\]
The three outer rectanges are obtained from distinguished squares by inverting some of the arrows, which are isomorphsims. By hypothesis, $F$ commutes around these squares. Our inductive hypothesis is that  $F$ has no monodromy around the central triangle. This clearly implies $F$ has no monodromy around the outer triangle, completing the proof.
\end{proof}

\subsection{Proof of Theorem~~\ref{thm:handleswap}}

Let us introduce some notation.  For the curves in the handleswap region, we write $_0$, $_0^H$, $_0$ and $_0^H$. We write $^H$ for $_u∪_0^H$, and similarly for $_0^H$. Hence, for the three diagrams appearing in the handleswap loop, we write
\[
\cH_1=(\Sigma\# \Sigma_0,\as,\bs), \quad \cH_2=(\Sigma\# \Sigma_0, \as^H, \bs),\quad \text{and} \quad \cH_3=(\Sigma\# \Sigma_0, \as^H, \bs^H).
\]

Similarly, let $_u$ be a collection of doubling curves on $Σ# Σ̅$. Let $_0$ be choice of doubling curves on $Σ_0# Σ̅_0$, as shown in Figure~\ref{fig:involutive-handleswap}. Write $$ for $_u∪_0$.

We write $$ for $$, decorated with $_u$. We pick a collection of four doubling curves $_0$ on $Σ_0# Σ̅_0$. We decorate $_1,$ $_2$, $_3$ with $$ to form the doubling enhanced diagrams $_1$, $_2$, and $_3$, respectively.

\begin{figure}[ht]
\centering
\input{handleswap-colortest.pdf_tex}
\caption{The double of the handleswap region. The left side is $\Sigma$, and the right is $\bar \Sigma$.}
\label{fig:involutive-handleswap}
\end{figure}

We note that for $i∈{1,2,3}$, there are canonical isomorphisms
\begin{equation}
\CFI(\frD)\iso \CFI(\frD_i), \label{eq:canonical-iso}
\end{equation}
given by $↦x×c_i,$
extended $[U,Q]/Q^2$-equivariantly. Here, we write $c_i$ for the unique intersection points
\[
\ve{c}_1\in \bT_{\a_0}\cap \bT_{\b_0}, \quad  \ve{c}_2\in \bT_{\a_0^H}\cap \bT_{\b_0}, \quad \text{and} \quad \ve{c}_3\in \bT_{\a_0^H}\cap \bT_{\b_0^H}.
\]

In fact, we have the following:
\begin{lem}
\label{lem:admissibility-handleswap-1}
The genus 2 diagram shown in Figure~\ref{fig:involutive-handleswap} is weakly admissible. Furthermore, each of the following subdiagrams has vanishing differential.
\begin{enumerate}
\item $(\Sigma_0\# \bar \Sigma_0,\bs_0\bar \bs_0, \Ds_0,w)$
\item $(\Sigma_0\# \bar \Sigma_0,\bs^H_0\bar \bs^H_0, \Ds_0,w)$.
\item $(\Sigma_0\# \bar \Sigma_0,\Ds_0, \as_0 \bar \as_0,w)$.
\item $(\Sigma_0\# \bar \Sigma_0,\Ds_0, \as^H_0\bar \as^H_0,w)$.
\item $(\Sigma_0\# \bar \Sigma_0,\as_0 \bar \bs_0, \Ds_0,w)$.
\item $(\Sigma_0\# \bar \Sigma_0,\as^H_0 \bar \bs^H_0, \Ds_0,w)$.
\end{enumerate}
\end{lem}
\begin{proof} Admissibility is a straightforward computation which we leave to the reader. To check that each of the stated complexes has vanishing differential, it suffices to check that each has the same number of generators as its homology. The first four subdiagrams represent a complex for $\#^2 S^1\times S^2$ and also have exactly four generators. Similarly the final two subdiagrams represent $S^3$, and each have a unique intersection point.
\end{proof}

By definition,
\[
\CFI(e)=\Psi_{\frD_1\to \frD_2}\quad \text{and} \quad \CFI(f)=\Psi_{\frD_2\to \frD_3},
\]
where $Ψ__i→_i+1$ is obtained by compressing a hyperbox as in Figure~\ref{def:transition-map-elementary-handleslide}. Computing the composition $Ψ__2→_3∘Ψ__1→_2$ is somewhat inefficient, however, since the involutive transition maps naturally allow for changing both the alpha and the beta curves simultaneously. Instead, we will consider the map $Ψ__1→_3$ obtained by by compressing the hyperbox in Figure~\ref{def:transition-map-elementary-handleslide}. (See Equation~\eqref{eq:Psi_1->3}, below). By Theorem~\ref{thm:naturality-fixed-Sigma},
\begin{equation}
\CFI(f)\circ \CFI(e)=\Psi_{\frD_2\to \frD_3}\circ \Psi_{\frD_1\to \frD_2}\simeq \Psi_{\frD_1\to \frD_3}.
\label{eq:f-e-composed}
\end{equation}

To prove Theorem~\ref{thm:handleswap}, we also need to understand the diffeomorphism map $(g)$. We note that the diffeomorphism $g$ does not fix $_0$. Hence,  by definition, this map is the composition
\begin{equation}
\CFI(g):=\Psi_{g(\frD_3)\to \frD_1}\circ T_g,
\label{eq:CFI-g-def}
\end{equation}
which we explain presently. The map $T_g$ is the tautological map which sends an intersection point $∈__0^H∩_̱̱_0^H$ to its image in $__0∩_̱̱_0$ under $g$, extended $[U,Q]/Q^2$-equivariantly. Also, $g(_3)$ is obtained by enhancing the Heegaard diagram $_1$ with the doubling curves $_u∪_0'$, where $_0'=g(_0)$. The map $Ψ_g(_3)→_1$ is built by compressing a hyperbox similar to the one in Figure~\ref{def:transition-map-elementary-handleslide} which realizes the change of doubling curves.

With respect to the canonical isomorphisms of~\eqref{eq:canonical-iso}, the map $T_g$ is clearly equal to the identity map.  Theorem~\ref{thm:handleswap} follows from  ~\eqref{eq:f-e-composed} and~\eqref{eq:CFI-g-def} and the following result.
\begin{prop}\label{prop:handleswap-factors=identity}
With respect to the canonical isomorphisms in~\eqref{eq:canonical-iso}, the maps $\Psi_{\frD_1\to \frD_3}$ and $\Psi_{g(\frD_3)\to \frD_1}$ are chain homotopic to the identity map.
\end{prop}

\begin{proof}
We focus first on the computation of $\Psi_{\frD_1\to \frD_3}$. By definition, this map is obtained by compressing the following hyperbox:
\begin{equation}
\begin{tikzcd}[labels=description, row sep=1cm]
\CF(\as,\bs)
\ar[r]
\ar[d, "F_{1}^{\bar{\b},\bar{\b}}"]
& \CF(\as^H,\bs)
\ar[r]
\ar[d, "F_{1}^{\bar{\b}^H,\bar{\b}}"]
& \CF(\as^H,\bs^H)
\ar[d,"F_{1}^{\bar{\b}^H,\bar{\b}^H}"]
\\
\CF(\as \bar{\bs}, \bs \bar{\bs})
\ar[r]
\ar[d]
\ar[dr,dashed]
& \CF(\as^H\bar{\bs}^H, \bs \bar{\bs})
\ar[r]
\ar[d]
\ar[dr,dashed]
& \CF(\as^H \bar{\bs}^H, \bs^H \bar{\bs}^H)
\ar[d]
\\
\CF(\as \bar \bs,\Ds)
\ar[r]
\ar[d]
\ar[dr,dashed]
& \CF(\as^H\bar \bs^H,\Ds)
\ar[r]
\ar[d]
\ar[dr,dashed]
& \CF(\as^H\bar \bs^H,\Ds)
\ar[d]
\\
\CF(\as\bar \bs, \as \bar \as)
\ar[r]
\ar[d, "F_3^{\a,\a }"]
\CF(\as^H \bar \bs^H, \as \bar \as)
\ar[r]
\ar[d, "F_3^{\a^H,\a }"]
\CF(\as^H \bar{\bs}^H, \as^H \bar{\as}^H)
\ar[d, "F_3^{\a^H,\a^H}"]
\\
\CF(\bar{\bs}, \bar{\as})
\ar[r]
\ar[drr,dashed]
\ar[d]
& \CF(\bar{\bs}^H,\bar{\as})
\ar[r]
& \CF(\bar{\bs}^H,\bar{\as}^H)
\ar[d]
\\
\CF(\bar \bs,\bar \as)
\ar[r]
\CF(\bar \bs, \bar \as^H)
\ar[r]
\CF(\bar \bs^H,\bar \as^H)
\end{tikzcd}
\label{eq:Psi_1->3}
\end{equation}
We note that in the above diagram, each polygon counting map occurs on a connected sum. One summand is $\Sigma$ or $\Sigma\# \bar \Sigma$ while the other summand is the genus 2 subsurface $\Sigma_0$, or its double. We will argue by viewing the diagram as a multi-stabilization and applying Lemma~\ref{lem:admissibility-handleswap-1} and Proposition~\ref{prop:multi-stabilization-counts}.

There is a subtlety in that the almost complex structures used to  construct the transition map Equation~\eqref{eq:Psi_1->3} are not naturally adapted to the genus four stabilization region where the handleswap occurs. Instead, we must first argue that we can change the above composition to one which is naturally adapted to the handleswap. To do so, we consider the closed curves on $\Sigma\# \bar \Sigma$ shown in Figure~\ref{fig:12}. We label these as $c_+,$ $c_-$, $c_{\delta}$ and $c_0$. These are as follows:
\begin{enumerate}
\item $c_0$ is a meridian of the connected sum tube used in the doubling operation.
\item $c_+$ is a closed curve on $\Sigma$ which bounds the handleswap region and the connected sum point.
\item $c_-$ is a symmetric closed curve on $\bar \Sigma$, and bounds the handleswap region as well as a the connected sum point.
\item $c_\delta$ is a closed curve on $\Sigma\# \bar \Sigma$ which bounds the union of both handleswap regions. Also, $c_{\delta}$ is disjoint from all $\Ds$ curves.
\end{enumerate}

\begin{figure}[ht]
\centering
\input{fig12.pdf_tex}
\caption{Curves along which we degenerate the almost complex structure.}
\label{fig:12}
\end{figure}

Write $J_{c_0}$ for an almost complex structure or family of almost complex structures which is nodal along $c_0$, and use similar notation to define almost complex structures degenerated on another curve or collections of pairwise disjoint curves on $\Sigma\# \bar \Sigma$.

We introduce some shorthand. Write $\CF(L_1),\dots, \CF(L_6)$ for the six horizontal levels of Equation~\eqref{eq:Psi_1->3}. (These are themselves 1-dimensional hyperboxes of chain complexes). By using change of almost complex structure hypercubes, similar to \cite{HHSZExact}*{Section~13}, we may assume that the complexes and maps in Equation~\eqref{eq:Psi_1->3} are computed with respect to almost complex structures which are \emph{maximally pinched} along $c_0$, $c_+$ and $c_-$, in the sense of \cite{HHSZExact}*{Section~8.3}. This gives the map $\Psi_{\frD_1\to \frD_3}$, as defined in the earlier sections of the present paper. On the other hand, using similar techniques, we may use change of almost complex structure hypercubes to relate this model to one which is maximally pinched along $c_+$, $c_-$ and $c_\delta$. This gives a hyperbox which we schematically indicate by the following diagram:
\begin{equation}
\begin{tikzcd}[labels=description, row sep=.8cm, column sep=4cm]
\CF_{J_{c_+}}(L_1)
\ar[r, "\id"]
\ar[d, "F_{1}"]
& \CF_{J_{c_+}}(L_1)
\ar[d, "F_{1}"]
\\
\CF_{J_{c_+,c_-,c_0}}(L_2)
\ar[r,"\Psi_{J_{c_+,c_-,c_0}\to J_{c_+,c_-,c_\delta}}"]
\ar[d]
\ar[dr,dashed]
& \CF_{J_{c_+,c_-,c_\delta}}(L_2)
\ar[d]
\\
\CF_{J}(L_3)
\ar[r, "\Psi_{J\to J_{c_\delta}}"]
\ar[d]
\ar[dr,dashed]
& \CF_{J_{c_\delta}}(L_3)
\ar[d]
\\
\CF_{J_{c_+,c_-,c_0}}(L_4)
\ar[r, "\Psi_{J_{c_+,c_-,c_0}\to J_{c_+,c_-,c_\delta}}"]
\ar[d, "F_3"]
\CF_{J_{c_+,c_-,c_\delta}}(L_4)
\ar[d, "F_3"]
\\