$\mathcal{A}_{\alpha={\rm hyp}}$ constructed from the unique hyperbolic
structure on $M$. Locally, the flat connection can be written as

$\displaystyle\mathcal{A}_{\alpha={\rm hyp}}=\omega+ie\;.$ (128)

Here $\omega$ and $e$ are the spin-connection and the dreibein, respectively,
constructed from the hyperbolic structure. Both can be considered as
$SO(3)$-valued 1-forms on $M$. The above complex combination gives a
$SL(2,\mathbb{C})$ connection which is actually the flat connection satisfying
$d\mathcal{A}+\mathcal{A}\wedge\mathcal{A}=0$. The flat connection gives the
most exponentially dominant contribution to the superconformal index
$\mathcal{I}_{\rm sci}(x)$ in the limit $x\rightarrow 1$ Bobev:2019zmz ;
Benini:2019dyp :

$\displaystyle\log\mathcal{I}^{UV}_{\rm sci}(x)\xrightarrow{\quad
x=e^{-\nu};\;\nu\rightarrow
i0^{-}\quad}-\frac{2i}{\nu}\textrm{vol}(M)+o(\nu^{0})\;.$

The above follows from (124) and (126) combined with following fact

$\displaystyle\textrm{Im}\left[CS[\mathcal{A}_{\alpha=\textrm{\rm
hyp}}]\right]=-\frac{1}{4\pi^{2}}\textrm{vol}(M)\;.$ (129)

Here $\textrm{vol}(M)$ is the hyperbolic volume of the hyperboilc 3-manifold.
The non-triviality of the superconformal index guarantees that the
$\mathcal{T}[M]$ flows to a non-trivial superconformal field theory. More
generally, from the same argument, one expects that

$\displaystyle\begin{split}&\textrm{If there is an irreducible
$SL(2,\mathbb{C})$ flat connection $\rho$ on $M$}\\\ &\textrm{with
$\textrm{Im}[CS[\rho]]\neq 0$}\\\ &\Rightarrow\\\ &\textrm{$\mathcal{T}[M]$
flows to a non-trivial superconformal field theory}.\end{split}$

####  $M$ satisfying conditions in (3) $\Rightarrow\
B_{M}^{\overline{\alpha}}(x)=B_{M}^{\alpha}(x)$

For a $SL(2,\mathbb{R})$ or $SU(2)$ flat connection $\mathcal{A}_{\alpha}$,
its complex conjugation $(\mathcal{A}_{\alpha})^{*}$ equals to itself up to a
transpose or sign

$\displaystyle(\mathcal{A}_{\alpha})^{*}=\mathcal{A}_{\alpha}\;,\quad\textrm{for
$SL(2,\mathbb{R})$ flat connection}\;,$
$\displaystyle(i\mathcal{A}_{\alpha})^{*}=(i\mathcal{A}_{\alpha})^{T}\;,\quad\textrm{for
$SU(2)$ flat connection}\;.$

This implies that two flat connections $\mathcal{A}_{\alpha}$ and
$(\mathcal{A}_{\alpha})^{*}$ have the same perturbative expansion in (115),
i.e. $F_{n}^{\alpha}=F^{\overline{\alpha}}_{n}=(F_{n}^{\alpha})^{*}$ for all
$n\geq 0$. Therefore, one finds that for a 3-manifold subject to the
conditions (3),

$\displaystyle\begin{split}&B_{M}^{\overline{\alpha}}(x)=B_{M}^{\alpha}(x)\;\;\textrm{for
all irreducible flat connections $\rho_{\alpha}$}\;,\\\ &\Rightarrow\
\mathcal{I}^{UV}_{\rm sci}(x)=\mathcal{I}^{UV}_{\rm top}(x)\textrm{ from
\eqref{3d-3d relation}}\;.\end{split}$

##  Appendix E Irreducible flat connections on
$M=S^{2}(3,3,-4/3),S^{2}(3,3,-5/3)$ and $S^{2}(2,3,8)$

To obtain irreducible $SL(2,\mathbb{C})$ flat connections on
$S^{2}(\frac{p_{1}}{q_{1}},\frac{p_{2}}{q_{2}},\frac{p_{3}}{q_{3}})$, we start
from following ansatz for their holonomy matrices

$\displaystyle\rho(x_{1})=\left(\begin{array}[]{cc}m&\epsilon\\\ 0&m^{-1}\\\
\end{array}\right)\bigg{|}_{\epsilon\in\\{0,1\\}}\;,\quad\rho(x_{2})=\left(\begin{array}[]{cc}a&b\\\
c&d\\\ \end{array}\right)\;,$ (134)
$\displaystyle\rho(x_{3})=\left(\rho(x_{1})\cdot\rho(x_{2})\right)^{-1}\;,\quad\rho(h)\in\\{+\mathbf{1},-\mathbf{1}\\}\;.$
(135)

Then, we solve the matrix equations for the holonomy matrices constrained from
the relations (47) in the fundamental group. Among those solutions, we need to
discard reducible flat connections for which all holonomy matrices are
mutually commuting, and choose gauge inequivalent sets of the irreducible
ones.

####  $M=S^{2}(3,3,-4/3)$

There are 3 irreducible flat connections $\rho_{\alpha=0,1,2}$ on
$M=S^{2}(3,3,-4/3)$ whose $SL(2,\mathbb{C})$ holonomy matrices are

$\displaystyle\begin{split}&\alpha=0:\\\
&\rho_{\alpha}(x_{1})=\left(\begin{array}[]{cc}e^{\frac{\pi i}{3}}&0\\\
0&e^{-\frac{\pi i}{3}}\\\
\end{array}\right),\;\rho_{\alpha=0}(h)=\left(\begin{array}[]{cc}-1&0\\\
0&-1\\\ \end{array}\right),\\\
&\rho_{\alpha}(x_{2})=\left(\begin{array}[]{cc}\frac{1}{2}+\frac{i}{6}\left(\sqrt{3}-2\sqrt{6}\right)&-\frac{\sqrt{2}}{3}\\\
1&\frac{1}{2}-\frac{i}{6}\left(\sqrt{3}-2\sqrt{6}\right)\\\
\end{array}\right),\\\
&\rho_{\alpha}(x_{3})=\left(\begin{array}[]{cc}\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{3}}+\frac{i}{\sqrt{6}}&\frac{1+i\sqrt{3}}{3\sqrt{2}}\\\
\frac{1}{2}i\left(\sqrt{3}+i\right)&\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{3}}-\frac{i}{\sqrt{6}}\\\
\end{array}\right),\\\ &\alpha=1:\\\
&\rho_{\alpha}(x_{1})=\left(\begin{array}[]{cc}e^{-\frac{2\pi i}{3}}&0\\\
0&e^{\frac{2\pi i}{3}}\\\
\end{array}\right),\;\rho_{\alpha}(x_{2})=\left(\begin{array}[]{cc}\frac{e^{-\frac{5i\pi}{6}}}{\sqrt{3}}&-\frac{2}{3}\\\
1&\frac{e^{\frac{5i\pi}{6}}}{\sqrt{3}}\\\ \end{array}\right),\\\
&\rho_{\alpha}(x_{3})=\left(\begin{array}[]{cc}-\frac{i}{\sqrt{3}}&\frac{2}{3}e^{\frac{4i\pi}{3}}\\\
e^{-\frac{i\pi}{3}}&\frac{i}{\sqrt{3}}\\\
\end{array}\right),\;\rho_{\alpha}(h)=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right),\\\ &\alpha=2:\\\
&\rho_{\alpha}(x_{1})=\left(\begin{array}[]{cc}e^{\frac{\pi i}{3}}&0\\\
0&e^{-\frac{\pi i}{3}}\\\
\end{array}\right),\;\rho_{\alpha}(h)=\left(\begin{array}[]{cc}-1&0\\\ 0&-1\\\
\end{array}\right),\\\
&\rho_{\alpha}(x_{2})=\left(\begin{array}[]{cc}\frac{1}{2}+\frac{i}{6}\left(\sqrt{3}+2\sqrt{6}\right)&\frac{\sqrt{2}}{3}\\\
1&\frac{1}{2}-\frac{i}{6}\left(\sqrt{3}+2\sqrt{6}\right)\\\
\end{array}\right),\\\
&\rho_{\alpha}(x_{3})=\left(\begin{array}[]{cc}-\frac{1}{\sqrt{2}}-\frac{i}{\sqrt{3}}-\frac{i}{\sqrt{6}}&-\frac{i\left(\sqrt{3}-i\right)}{3\sqrt{2}}\\\
\frac{1}{2}i\left(\sqrt{3}+i\right)&-\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{3}}+\frac{i}{\sqrt{6}}\\\
\end{array}\right).\end{split}$ (136)

One can confirm that all of them are conjugate to either $SU(2)$ or
$SL(2,\mathbb{R})$ flat connections by checking that
$\textrm{Tr}(\rho(a))\in\mathbb{R}$ for all $a\in\pi_{1}(M)$.
$\rho_{\alpha=2}$ is conjugate to $SL(2,\mathbb{R})$ flat connection while the
other twos are conjugate to $SU(2)$ flat connections. Thus the 3-manifold $M$
satisfies the topological conditions in (3). The eigenvalues of the holonomy
matrices are

$\displaystyle\begin{split}&\alpha=0\;:\;n_{\alpha 1}=\frac{1}{2}\;,n_{\alpha
2}=\frac{1}{2}\;,n_{\alpha 3}=\frac{1}{2}\;,\lambda_{\alpha}=\frac{1}{2}\;,\\\
&\alpha=1\;:\;n_{\alpha 1}=1\;,n_{\alpha 2}=1\;,n_{\alpha
3}=\frac{1}{2}\;,\lambda_{\alpha}=0\;,\\\ &\alpha=2\;:\;n_{\alpha
1}=\frac{1}{2}\;,n_{\alpha 2}=\frac{1}{2}\;,n_{\alpha
3}=\frac{3}{2}\;,\lambda_{\alpha}=\frac{1}{2}\;.\end{split}$ (137)

See eq. (V.1) for the definition of the above $n_{\alpha i}$ and
$\lambda_{\alpha}$. From this result, we obtain $CS[\rho_{\alpha}]$ and
$\textrm{Tor}[\rho_{\alpha}]$ given in (50) using the formula in (49)

####  $M=S^{2}(3,3,-5/3)$

There are 4 irreducible flat connections $\rho_{\alpha=0,1,2,3}$ on
$M=S^{2}(3,3,-5/3)$ whose $SL(2,\mathbb{C})$ holonomy matrices are

$\displaystyle\begin{split}&\alpha=0:\\\
&\rho_{\alpha}(x_{1})=\left(\begin{array}[]{cc}e^{-\frac{2\pi i}{3}}&0\\\
0&e^{\frac{2\pi i}{3}}\\\
\end{array}\right),\;\rho_{\alpha=0}(h)=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right),\\\
&\rho_{\alpha}(x_{2})=\left(\begin{array}[]{cc}-\frac{1}{2}-\frac{i\left(\sqrt{5}+2\right)}{2\sqrt{3}}&\frac{\sqrt{5}}{3}\\\
1&-\frac{1}{2}+i\sqrt{\frac{\sqrt{5}}{3}+\frac{3}{4}}\\\
\end{array}\right),\\\
&\rho_{\alpha}(x_{3})=\left(\begin{array}[]{cc}u&-\frac{i\left(\sqrt{3}-i\right)}{3\sqrt{2}}\\\
\frac{1}{2}i\left(\sqrt{3}+i\right)&u^{*}\\\ \end{array}\right),\\\
&\alpha=1:\\\ &\rho_{\alpha}(x_{1})=\left(\begin{array}[]{cc}e^{\frac{\pi
i}{3}}&0\\\ 0&e^{-\frac{\pi i}{3}}\\\
\end{array}\right),\;\rho_{\alpha}(h)=\left(\begin{array}[]{cc}-1&0\\\ 0&-1\\\
\end{array}\right),\\\
&\rho_{\alpha}(x_{2})=\left(\begin{array}[]{cc}\frac{1}{6}\left(3-i\sqrt{15}\right)&-\frac{1}{3}\\\
1&\frac{1}{6}\left(3+i\sqrt{15}\right)\\\ \end{array}\right),\\\
&\rho_{\alpha}(x_{3})=\left(\begin{array}[]{cc}\frac{1}{6}\sqrt[6]{-1}\left(\sqrt{15}-3i\right)&\frac{\sqrt[3]{-1}}{3}\\\
(-1)^{2/3}&\frac{1}{6}\sqrt[3]{-1}\left(3-i\sqrt{15}\right)\\\
\end{array}\right)\\\ &\alpha=2:\\\
&\rho_{\alpha}(x_{1})=\left(\begin{array}[]{cc}e^{-\frac{2\pi i}{3}}&0\\\
0&e^{\frac{2\pi i}{3}}\\\
\end{array}\right),\;\rho_{\alpha}(h)=\left(\begin{array}[]{cc}1&0\\\ 0&1\\\
\end{array}\right),\\\
&\rho_{\alpha}(x_{2})=\left(\begin{array}[]{cc}-\frac{1}{2}-\frac{i\left(-\sqrt{5}+2\right)}{2\sqrt{3}}&-\frac{\sqrt{5}}{3}\\\
1&-\frac{1}{2}+i\sqrt{\frac{-\sqrt{5}}{3}+\frac{3}{4}}\\\
\end{array}\right)\;,\\\
&\rho_{\alpha}(x_{3})=\left(\begin{array}[]{cc}v&-\frac{1}{3}\sqrt[3]{-1}\sqrt{5}\\\
-(-1)^{2/3}&v^{*}\\\ \end{array}\right),\\\ &\alpha=3:\\\
&\rho_{\alpha}(x_{1})=\left(\begin{array}[]{cc}e^{\frac{\pi i}{3}}&0\\\
0&e^{-\frac{\pi i}{3}}\\\
\end{array}\right),\;\rho_{\alpha}(h)=\left(\begin{array}[]{cc}-1&0\\\ 0&-1\\\
\end{array}\right),\\\
&\rho_{\alpha}(x_{2})=\left(\begin{array}[]{cc}\frac{1}{6}\left(3+i\sqrt{15}\right)&-\frac{1}{3}\\\
1&\frac{1}{6}\left(3-i\sqrt{15}\right)\\\ \end{array}\right),\\\
&\rho_{\alpha}(x_{3})=\left(\begin{array}[]{cc}\frac{1}{6}\sqrt[6]{-1}\left(-\sqrt{15}-3i\right)&\frac{\sqrt[3]{-1}}{3}\\\
(-1)^{2/3}&\frac{1}{6}\sqrt[3]{-1}\left(3+i\sqrt{15}\right)\\\
\end{array}\right)\;.\end{split}$ (138)

Here $\\{u\simeq-0.809017-1.04444i,\;v\simeq
0.309017-0.398939i,\;u^{*},\;v^{*}\\}$ are the solutions to an algebraic
equation $9x^{4}+9x^{3}+9x^{2}-6x+4=0$. $\rho_{\alpha=0}$ is conjugate to an
$SL(2,\mathbb{R})$ flat connection while the other threes are conjugate to
$SU(2)$ flat connections.

####  $M=S^{2}(2,3,8)$

This manifold has a non-trivial $H_{1}(M,\mathbb{Z}_{2})=\mathbb{Z}_{2}$.
Thus, the set of $PSL(2,\mathbb{C})$ flat connections is different from the
set of $SL(2,\mathbb{C})$ flat connections. There are four irreducible
$PSL(2,\mathbb{C})$ flat connections
$\\{\rho^{PSL}_{\alpha}\\}_{\alpha=0}^{3}$ whose holonomy matrices are
$\rho_{\alpha}^{PSL}(a)=[\rho^{SL}(a)]$ with

$\displaystyle\begin{split}&\rho^{SL}_{\alpha}(x_{1})=\left(\begin{array}[]{cc}-i&0\\\
0&i\\\
\end{array}\right)\;,\quad\rho^{SL}_{\alpha}(h)=\left(\begin{array}[]{cc}-1&0\\\
0&-1\\\ \end{array}\right)\;,\\\
&\rho^{SL}_{\alpha=0}(x_{2})=\left(\begin{array}[]{cc}\frac{1}{2}\left(1-i\sqrt{\sqrt{2}+2}\right)&\frac{1}{4}\left(\sqrt{2}-1\right)\\\
1&\frac{1}{2}\left(1+i\sqrt{\sqrt{2}+2}\right)\\\ \end{array}\right)\;,\\\
&\rho^{SL}_{\alpha=0}(x_{3})=\left(\begin{array}[]{cc}\frac{1}{2}\left(-\sqrt{\sqrt{2}+2}+i\right)&\frac{1}{4}i\left(\sqrt{2}-1\right)\\\
-i&\frac{1}{2}\left(-\sqrt{\sqrt{2}+2}-i\right)\\\ \end{array}\right)\;,\\\
&\rho^{SL}_{\alpha=1}(x_{2})=\left(\begin{array}[]{cc}\frac{1}{2}-\frac{i}{\sqrt{2}}&-\frac{1}{4}\\\
1&\frac{1}{2}+\frac{i}{\sqrt{2}}\\\ \end{array}\right)\;,\\\
&\rho^{SL}_{\alpha=1}(x_{3})=\left(\begin{array}[]{cc}-\frac{1}{\sqrt{2}}+\frac{i}{2}&-\frac{i}{4}\\\
-i&-\frac{1}{\sqrt{2}}-\frac{i}{2}\\\ \end{array}\right)\;,\\\
&\rho^{SL}_{\alpha=2}(x_{2})=\left(\begin{array}[]{cc}\frac{1}{2}\left(1-i\sqrt{2-\sqrt{2}}\right)&\frac{1}{4}\left(-\sqrt{2}-1\right)\\\
1&\frac{1}{2}\left(1+i\sqrt{2-\sqrt{2}}\right)\\\ \end{array}\right)\;,\\\
&\rho^{SL}_{\alpha=2}(x_{3})=\left(\begin{array}[]{cc}\frac{1}{2}\left(-\sqrt{2-\sqrt{2}}+i\right)&-\frac{1}{4}i\left(\sqrt{2}+1\right)\\\
-i&\frac{1}{2}\left(-\sqrt{2-\sqrt{2}}-i\right)\\\ \end{array}\right)\;,\\\
&\rho^{SL}_{\alpha=3}(x_{2})=\left(\begin{array}[]{cc}\frac{1}{2}&-\frac{3}{4}\\\
1&\frac{1}{2}\\\
\end{array}\right)\;,\quad\rho^{SL}_{\alpha=3}(x_{3})=\left(\begin{array}[]{cc}\frac{i}{2}&-\frac{3i}{4}\\\
-i&-\frac{i}{2}\\\ \end{array}\right)\;.\end{split}$ (139)

There are two $\mathbb{Z}_{2}$ flat connections,
$H^{1}(M,\mathbb{Z}_{2})=\\{1,\eta\\}$, where

$\displaystyle\eta(x_{1})=\eta(x_{3})=-1\;,\quad\eta(x_{2})=\eta(h)=0\;.$
(140)

Since the $\mathop{\mathrm{tr}}(\rho^{PSL}_{\alpha=0,1,2}(x_{3}))\neq 0$, the
first three $PSL(2,\mathbb{C})$ flat connections, $\rho_{\alpha=0,1,2}$, are
not invariant under the tensoring with $\eta$. On the other hand, the last
flat connection $\rho_{\alpha=3}$ is invariant under the tensoring. Among the
4 $PSL(2,\mathbb{C})$ flat connections, $\rho^{PSL}_{\alpha=0,2}$ can be
uplifted to $SL(2,\mathbb{C})$ flat connections $\rho_{\alpha=0,2}$ whose
holonomy matrices are $\rho^{SL}_{\alpha=0,2}$ given above. Taking into
account of tensoring with the $\mathbb{Z}_{2}$ flat connections, there are 4
irreducible $SL(2,\mathbb{C})$ flat connections, $\rho_{\alpha=0,2}$ and
$\eta\otimes\rho_{\alpha=0,2}$

##  Appendix F Modular structure of TFT$[M]$ in Table 1 and 3

Here we give modular structures of $\textrm{TFT}[M]$ in Table 1 and Table 3.
For simplicity of presentation, we only give explicit modular data for a
specific choice of true vacuum if there are several possible choices. See the
related discussion around the equation (22). The modular data of
$\textrm{TFT}[M]$ with other choices can be easily obtained from the Lorentz
symmetry fractionalization procedure or the parity operation.

###  F.1 Trivial $H_{1}(M,\mathbb{Z}_{2})$

#### $S^{2}\left(2,3,3\right)$

There is only one irreducible $SL(2,\mathbb{C})$ flat connection $\rho$ with

$\displaystyle\begin{split}&\textrm{Tor}[\rho]=\tfrac{1}{2}\;,\quad
CS[\rho]=-\tfrac{7}{24}\;.\end{split}$

Note that the central charge $c_{2d}=\pm 24CS[\rho]=0\;(\textrm{mod
}\frac{1}{2})$ and $\textrm{GSD}_{g}=(2\textrm{Tor}[\rho])^{g-1}=1$ for all
$g\geq 0$. The modular structure is identical to that of the trivial bosonic
topological phase $1^{B}_{1}$.

#### $S^{2}(2,3,5)$

There are two irreducible $SL(2,\mathbb{C})$ flat connections
$\rho_{\alpha=0,1}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{1}=\left\\{\tfrac{\left(5+\sqrt{5}\right)}{4},\tfrac{\left(5-\sqrt{5}\right)}{4}\right\\}\;,\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{1}=\left\\{\tfrac{41}{120},\tfrac{89}{120}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3},R=\textrm{Sym}^{\alpha}\Box\right)\;.$

The $S$-matrix and the topological spins
$h_{\alpha}=\pm(CS[\rho_{\alpha}]-CS[\rho_{\alpha=0}])$ are

$\displaystyle\begin{split}&S=\left(\begin{array}[]{cc}\sqrt{\frac{1}{10}\left(5-\sqrt{5}\right)}&\sqrt{\frac{1}{10}\left(5+\sqrt{5}\right)}\\\
\sqrt{\frac{1}{10}\left(\sqrt{5}+5\right)}&-\sqrt{\frac{1}{10}\left(5-\sqrt{5}\right)}\\\
\end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{1}=\pm\left\\{0,\tfrac{2}{5}\right\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\\{\textrm{GSD}_{g}\\}_{g=0,1,\ldots}=\left\\{1,2,5,15,50,175,\ldots\right\\}\;.$

The modular structure is identical to that of $2^{B}_{\pm 14/5}$.

#### $S^{2}\left(2,3,\frac{5}{2}\right)$

There are two irreducible $SL(2,\mathbb{C})$ flat connections
$\rho_{\alpha=0,1}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{1}=\left\\{\tfrac{\left(5-\sqrt{5}\right)}{4},\tfrac{\left(\sqrt{5}+5\right)}{4}\right\\}\;,\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{1}=\left\\{\tfrac{83}{120},\tfrac{107}{120}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is non-unitary. The
flat-connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3}^{2},R=\textrm{Sym}^{\alpha}\Box\right)\;.$

The $S$-matrix and the topological spins
$h_{\alpha}=\pm(CS[\rho_{\alpha}]-CS[\rho_{\alpha=0}])$ are

$\displaystyle\begin{split}&S=\left(\begin{array}[]{cc}\sqrt{\frac{1}{10}\left(5+\sqrt{5}\right)}&-\sqrt{\frac{1}{10}\left(\sqrt{5}-5\right)}\\\
-\sqrt{\frac{1}{10}\left(\sqrt{5}-5\right)}&-\sqrt{\frac{1}{10}\left(5+\sqrt{5}\right)}\\\
\end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{1}=\pm\left\\{0,\tfrac{1}{5}\right\\}\end{split}$

The ground state degeneracy is

$\displaystyle\\{\textrm{GSD}_{g}\\}_{g=0,1,\ldots}=\left\\{1,2,5,15,50,175,\ldots\right\\}\;.$

The modular structure is identical to that of
$\textrm{Gal}_{\sharp=2}[(A_{1},3)_{1/2}]=(\textrm{Lee-Yang model})$.

#### $S^{2}(3,3,3)$

There are two irreducible flat connections $\rho_{\alpha=0,1}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{1}=\left\\{1,1\right\\}\;,\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{1}=\left\\{0,\tfrac{3}{4}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3},R=\textrm{Sym}^{\alpha}\Box\right)\;.$

The $S$-matrix and the topological spin $\\{h_{\alpha}\\}_{\alpha=0}^{1}$ are

$\displaystyle\begin{split}&S=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1&1\\\
1&-1\\\ \end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{1}=\pm\left\\{0,\tfrac{3}{4}\right\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\textrm{GSD}_{g}=2^{g}\;.$

The modular structure is identical to that of $2^{B}_{\pm 1}$.

#### $S^{2}(2,3,7)$

There are three irreducible flat $SL(2,\mathbb{C})$ connections
$\rho_{\alpha=0,1,2}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{2}=\tfrac{7}{8}\left\\{\csc^{2}\left(\tfrac{\pi}{7}\right),\csc^{2}\left(\tfrac{2\pi}{7}\right),\csc^{2}\left(\tfrac{3\pi}{7}\right)\right\\}\;,\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{2}=\left\\{\tfrac{127}{168},\tfrac{151}{168},\tfrac{79}{168}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3},R=\textrm{Sym}^{\alpha}\Box\right)$

The $S$-matrix and the topological spin $\\{h_{\alpha}\\}_{\alpha=0}^{2}$ are

$\displaystyle\begin{split}&S=\left(\begin{array}[]{ccc}\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}&\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}&\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}\\\
\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}&-\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}&\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}\\\
\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}&\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}&-\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}\\\
\end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{2}=\pm\\{0,\tfrac{1}{7},\tfrac{5}{7}\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,3,14,98,833,7546,\ldots\right\\}\;.$

The modular structure is identical to that of $3^{B}_{\pm 8/7}$.

#### $S^{2}\left(2,3,\frac{7}{2}\right)$

There are three irreducible $SL(2,\mathbb{C})$ flat connections
$\rho_{\alpha=0,1,2}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{2}=\tfrac{7}{8}\left\\{\sec^{2}\left(\tfrac{\pi}{14}\right),\csc^{2}\left(\tfrac{\pi}{7}\right),\sec^{2}\left(\tfrac{3\pi}{14}\right)\right\\}\;,\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{2}=\left\\{\tfrac{25}{168},\tfrac{121}{168},\tfrac{1}{168}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is non-unitary. The
flat-connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3}^{3},R=\textrm{Sym}^{\alpha}\Box\right)$

The $S$-matrix and the topological spins $h_{\alpha}$ are

$\displaystyle\begin{split}&S=\left(\begin{array}[]{ccc}\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}&\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}&-\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}\\\
\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}&\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}&\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}\\\
-\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}&\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}&-\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}\\\
\end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{2}=\pm\\{0,\tfrac{4}{7},\tfrac{6}{7}\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,3,14,98,833,7546,\ldots\right\\}\;.$

The modular structure is identical to that of
$\textrm{Gal}_{\sharp=2}[(A_{1},5)_{1/2}]$.

#### $S^{2}\left(2,3,\frac{7}{3}\right)$

There are three $SL(2,\mathbb{C})$ irreducible flat connections
$\rho_{\alpha=0,1,2}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{2}=\tfrac{7}{8}\left\\{\sec^{2}\left(\tfrac{3\pi}{14}\right),\sec^{2}\left(\tfrac{\pi}{14}\right),\csc^{2}\left(\tfrac{\pi}{7}\right)\right\\}\;,\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{2}=\left\\{\tfrac{19}{168},\tfrac{139}{168},\tfrac{115}{168}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is non-unitary. The
flat-connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3}^{2},R=\textrm{Sym}^{\alpha}\Box\right)\;.$

The $S$-matrix and the topological spins $h_{\alpha}$ are

$\displaystyle\begin{split}&S=\left(\begin{array}[]{ccc}\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}&-\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}&\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}\\\
-\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}&-\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}&\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}\\\
\frac{2\sin\left(\frac{\pi}{7}\right)}{\sqrt{7}}&\frac{2\cos\left(\frac{3\pi}{14}\right)}{\sqrt{7}}&\frac{2\cos\left(\frac{\pi}{14}\right)}{\sqrt{7}}\\\
\end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{2}=\pm\\{0,\tfrac{5}{7},\tfrac{4}{7}\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,3,14,98,833,7546,\ldots\right\\}\;.$

The modular structure is identical to that of
$\textrm{Gal}_{\sharp=3}[(A_{1},5)_{1/2}]$.

#### $S^{2}\left(3,\frac{3}{2},4\right)$

There are three irreducible flat connections $\rho_{\alpha=0,1,2}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{2,1,2\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{11}{16},\tfrac{3}{4},\tfrac{3}{16}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3},R=\textrm{Sym}^{\alpha}\Box\right)\;.$

The $S$-matrix and the topological spin $\\{h_{\alpha}\\}_{\alpha=0}^{2}$ are

$\displaystyle\begin{split}&S=\frac{2}{3}\left(\begin{array}[]{ccc}\frac{1}{2}&\frac{1}{\sqrt{2}}&\frac{1}{2}\\\
\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\\\
\frac{1}{2}&-\frac{1}{\sqrt{2}}&\frac{1}{2}\\\ \end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{2}=\pm\left\\{0,\tfrac{1}{16},\tfrac{1}{2}\right\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,3,10,36,136,528,\ldots\right\\}\;.$

The modular structure is identical to that of $3^{B}_{\pm 1/2}$.

####  $S^{2}\left(3,3,4\right)$

There are three irreducible flat connections $\rho_{\alpha=0,1,2}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{2,1,2\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{37}{48},\tfrac{1}{12},\tfrac{13}{48}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3},R=\textrm{Sym}^{\alpha}\Box\right)$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{2}$ are

$\displaystyle\begin{split}&S=\frac{2}{3}\left(\begin{array}[]{ccc}\frac{1}{2}&\frac{1}{\sqrt{2}}&\frac{1}{2}\\\
\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\\\
\frac{1}{2}&-\frac{1}{\sqrt{2}}&\frac{1}{2}\\\ \end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{2}=\pm\left\\{0,\tfrac{5}{16},\tfrac{1}{2}\right\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,3,10,36,136,528,\ldots\right\\}\;.$

The modular structure is identical to that of $3^{B}_{\pm 5/2}$.

####  $S^{2}(2,3,9)$

There are four irreducible flat connections $\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\tfrac{9}{8}\left\\{\csc^{2}\left(\tfrac{\pi}{9}\right),\csc^{2}\left(\tfrac{2\pi}{9}\right),\tfrac{4}{3},\sec^{2}\left(\tfrac{\pi}{18}\right)\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{55}{72},\tfrac{31}{72},\tfrac{13}{24},\tfrac{7}{72}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3},R=\textrm{Sym}^{\alpha}\Box\right)$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{3}$ are

$\displaystyle
S=\left(\begin{array}[]{cccc}\frac{2}{3}\sin\left(\frac{\pi}{9}\right)&\frac{2}{3}\sin\left(\frac{2\pi}{9}\right)&\frac{1}{\sqrt{3}}&\frac{2}{3}\cos\left(\frac{\pi}{18}\right)\\\
\frac{2}{3}\sin\left(\frac{2\pi}{9}\right)&-\frac{2}{3}\cos\left(\frac{\pi}{18}\right)&\frac{1}{\sqrt{3}}&-\frac{2}{3}\sin\left(\frac{\pi}{9}\right)\\\
\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&0&-\frac{1}{\sqrt{3}}\\\
\frac{2}{3}\cos\left(\frac{\pi}{18}\right)&-\frac{2}{3}\sin\left(\frac{\pi}{9}\right)&-\frac{1}{\sqrt{3}}&\frac{2}{3}\sin\left(\frac{2\pi}{9}\right)\\\
\end{array}\right),$ (145)
$\displaystyle\\{h_{\alpha}\\}_{\alpha=0}^{3}=\pm\left\\{0,\tfrac{2}{3},\tfrac{7}{9},\tfrac{1}{3}\right\\}\;.$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,30,414,7317,137862,\ldots\right\\}\;.$

The modular structure is identical to that of $4^{B}_{\pm 10/3}$.

#### $S^{2}(2,3,\frac{9}{2})$

There are four irreducible flat connections $\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\tfrac{9}{8}\left\\{\sec^{2}\left(\tfrac{\pi}{18}\right),\csc^{2}\left(\tfrac{\pi}{9}\right),\tfrac{4}{3},\csc^{2}\left(\tfrac{2\pi}{9}\right)\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{29}{72},\tfrac{53}{72},\tfrac{7}{24},\tfrac{5}{72}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is non-unitary. The
flat-connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3}^{5},R=\textrm{Sym}^{\alpha}\Box\right)\;.$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{3}$ are

$\displaystyle
S=\frac{2}{3}\left(\begin{array}[]{cccc}\cos\left(\frac{\pi}{18}\right)&-\sin\left(\frac{\pi}{9}\right)&-\frac{\sqrt{3}}{2}&\sin\left(\frac{2\pi}{9}\right)\\\
-\sin\left(\frac{\pi}{9}\right)&-\sin\left(\frac{2\pi}{9}\right)&-\frac{\sqrt{3}}{2}&-\cos\left(\frac{\pi}{18}\right)\\\
-\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}&0&\frac{\sqrt{3}}{2}\\\
\sin\left(\frac{2\pi}{9}\right)&-\cos\left(\frac{\pi}{18}\right)&\frac{\sqrt{3}}{2}&-\sin\left(\frac{\pi}{9}\right)\\\
\end{array}\right)\;,$ (150)
$\displaystyle\\{h_{\alpha}\\}_{\alpha=0}^{3}=\pm\left\\{0,\tfrac{1}{3},\tfrac{8}{9},\tfrac{2}{3}\right\\}\;.$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,30,414,7317,137862,\ldots\right\\}\;.$

The modular structure is identical to that of
$\textrm{Gal}_{\sharp=2}[(A_{1},7)_{1/2}]$.

####  $S^{2}(2,3,\frac{9}{4})$

There are four irreducible $SL(2,\mathbb{C})$ flat connections
$\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\tfrac{9}{8}\left\\{\csc^{2}\left(\tfrac{2\pi}{9}\right),\sec^{2}\left(\tfrac{\pi}{18}\right),\tfrac{4}{3},\csc^{2}\left(\tfrac{\pi}{9}\right)\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{25}{72},\tfrac{1}{72},\tfrac{19}{24},\tfrac{49}{72}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is non-unitary. The
flat-connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3}^{2},R=\textrm{Sym}^{\alpha}\Box\right)\;.$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{3}$ are

$\displaystyle
S=\frac{2}{3}\left(\begin{array}[]{cccc}\sin\left(\frac{2\pi}{9}\right)&-\cos\left(\frac{\pi}{18}\right)&\frac{\sqrt{3}}{2}&-\sin\left(\frac{\pi}{9}\right)\\\
-\cos\left(\frac{\pi}{18}\right)&\sin\left(\frac{\pi}{9}\right)&\frac{\sqrt{3}}{2}&-\sin\left(\frac{2\pi}{9}\right)\\\
\frac{\sqrt{3}}{2}&\frac{\sqrt{3}}{2}&0&-\frac{\sqrt{3}}{2}\\\
-\sin\left(\frac{\pi}{9}\right)&-\sin\left(\frac{2\pi}{9}\right)&-\frac{\sqrt{3}}{2}&-\cos\left(\frac{\pi}{18}\right)\\\
\end{array}\right)\;,$ (155)
$\displaystyle\\{h_{\alpha}\\}_{\alpha=0}^{3}=\pm\left\\{0,\tfrac{2}{3},\tfrac{4}{9},\tfrac{1}{3}\right\\}\;.$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,30,414,7317,137862,\ldots\right\\}\;.$

The modular structure is identical to that of
$\textrm{Gal}_{\sharp=4}[(A_{1},7)_{1/2}]$.

####  $S^{2}\left(3,3,5\right)$

There are four irreducible flat connections $\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\
\left\\{\tfrac{\sqrt{5}+5}{2},\tfrac{\sqrt{5}-5}{2},\tfrac{\sqrt{5}-5}{2},\tfrac{\sqrt{5}+5}{2}\right\\}\;,\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{47}{60},\tfrac{2}{15},\tfrac{23}{60},\tfrac{8}{15}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\rho_{\alpha}\rightarrow\left(a=x_{3},R=\textrm{Sym}^{\alpha}\Box\right)\;.$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{2}$ are

$\displaystyle\begin{split}&S=\frac{1}{2}\left(\begin{array}[]{cccc}\sqrt{1-\tfrac{1}{\sqrt{5}}}&\sqrt{1+\tfrac{1}{\sqrt{5}}}&\sqrt{1+\frac{1}{\sqrt{5}}}&\sqrt{1-\frac{1}{\sqrt{5}}}\\\
\sqrt{1+\frac{1}{\sqrt{5}}}&\sqrt{1-\frac{1}{\sqrt{5}}}&-\sqrt{1-\frac{1}{\sqrt{5}}}&-\sqrt{1+\frac{1}{\sqrt{5}}}\\\
\sqrt{1+\frac{1}{\sqrt{5}}}&-\sqrt{1-\frac{1}{\sqrt{5}}}&-\sqrt{1-\frac{1}{\sqrt{5}}}&\sqrt{1+\frac{1}{\sqrt{5}}}\\\
\sqrt{1-\frac{1}{\sqrt{5}}}&-\sqrt{1+\frac{1}{\sqrt{5}}}&\sqrt{1+\frac{1}{\sqrt{5}}}&-\sqrt{1-\frac{1}{\sqrt{5}}}\\\
\end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{2}=\pm\left\\{0,\tfrac{7}{20},\tfrac{3}{5},\tfrac{3}{4}\right\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,20,120,800,5600,\ldots\right\\}\;.$

The modular structure is identical to that of $4^{B}_{\pm 19/5}$.

#### $S^{2}\left(\frac{3}{2},3,\frac{5}{2}\right)$

There are four irreducible flat connections $\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\
\left\\{\tfrac{\sqrt{5}-5}{2},\tfrac{\sqrt{5}+5}{2},\tfrac{\sqrt{5}+5}{2},\tfrac{\sqrt{5}-5}{2}\right\\}\;,\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{3}{5},\tfrac{2}{5},\tfrac{17}{20},\tfrac{13}{20}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is non-unitary. The
flat-connection-to-loop operator map is

$\displaystyle\rho_{\alpha=1}\rightarrow\left(a=x_{3}^{2},R=\Box\right)\;,\quad\rho_{\alpha=2}\rightarrow\left(a=x_{2},R=\Box\right)\;,$
$\displaystyle\rho_{\alpha=3}\rightarrow\left(a=x_{3}^{2},R=\Box\right)\otimes\left(a=x_{2},R=\Box\right)\;.$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{3}$ are

$\displaystyle\begin{split}&S=\frac{1}{2}\left(\begin{array}[]{cccc}\sqrt{1+\frac{1}{\sqrt{5}}}&-\sqrt{1-\frac{1}{\sqrt{5}}}&\sqrt{1+\frac{1}{\sqrt{5}}}&-\sqrt{1-\frac{1}{\sqrt{5}}}\\\
-\sqrt{1-\frac{1}{\sqrt{5}}}&-\sqrt{1+\frac{1}{\sqrt{5}}}&-\sqrt{1-\frac{1}{\sqrt{5}}}&-\sqrt{1+\frac{1}{\sqrt{5}}}\\\
\sqrt{1+\frac{1}{\sqrt{5}}}&-\sqrt{1-\frac{1}{\sqrt{5}}}&-\sqrt{1+\frac{1}{\sqrt{5}}}&\sqrt{1-\frac{1}{\sqrt{5}}}\\\
-\sqrt{1-\frac{1}{\sqrt{5}}}&-\sqrt{1+\frac{1}{\sqrt{5}}}&\sqrt{1-\frac{1}{\sqrt{5}}}&\sqrt{1+\frac{1}{\sqrt{5}}}\\\
\end{array}\right)\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{3}=\pm\left\\{0,\tfrac{4}{5},\tfrac{1}{4},\tfrac{1}{20}\right\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,20,120,800,5600,\ldots\right\\}\;.$

The modular structure is identical to that of $\left(\textrm{Gal}_{\sharp
2}[(A_{1},3)_{1/2}]\right)\otimes(2^{B}_{\pm 1})$.

#### $S^{2}\left(2,5,5\right)$

There are four irreducible flat connections $\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{5(3+\sqrt{5})}{4},\tfrac{5}{2},\tfrac{5}{2},\tfrac{5(3-\sqrt{5})}{4}\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{31}{40},\tfrac{3}{8},\tfrac{3}{8},\tfrac{39}{40}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\rho_{\alpha=1}\rightarrow\left(a=x_{2},R=\Box\right)\;,\quad\rho_{\alpha=2}\rightarrow\left(a=x_{3},R=\Box\right)\;,$
$\displaystyle\rho_{\alpha=3}\rightarrow\left(a=x_{2},R=\Box\right)\otimes\left(a=x_{3},R=\Box\right)\;.$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{2}$ are

$\displaystyle
S=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}\tfrac{\left(\sqrt{5}-1\right)}{2}&1&1&\frac{\left(\sqrt{5}+1\right)}{2}\\\
1&\frac{\left(1-\sqrt{5}\right)}{2}&\frac{\left(\sqrt{5}+1\right)}{2}&-1\\\
1&\frac{\left(\sqrt{5}+1\right)}{2}&\frac{\left(1-\sqrt{5}\right)}{2}&-1\\\
\frac{\left(\sqrt{5}+1\right)}{2}&-1&-1&\frac{\left(\sqrt{5}-1\right)}{2}\\\
\end{array}\right)\;\;,$ (160)
$\displaystyle\\{h_{\alpha}\\}_{\alpha=0}^{3}=\pm\left\\{0,\tfrac{3}{5},\tfrac{3}{5},\tfrac{1}{5}\right\\}\;.$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,25,225,2500,30625,\ldots\right\\}\;.$

The modular structure is identical to that of $4^{B}_{\pm 12/5}$.

####  $S^{2}\left(2,4,\frac{5}{4}\right)$

There are four irreducible flat connections $\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{5(3+\sqrt{5})}{4},\tfrac{5}{2},\tfrac{5}{2},\tfrac{5(3-\sqrt{5})}{4}\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{5}{8},\tfrac{9}{40},\tfrac{1}{40},\tfrac{5}{8}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is unitary. The flat-
connection-to-loop operator map is

$\displaystyle\begin{split}&\rho_{\alpha=1}\rightarrow\left(a=x_{2},R=\Box\right)\;,\quad\rho_{\alpha=2}\rightarrow\left(a=x_{3},R=\Box\right)\;,\\\
&\rho_{\alpha=3}\rightarrow\left(a=x_{2},R=\Box\right)\otimes\left(a=x_{3},R=\Box\right)\;.\end{split}$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{3}$ are

$\displaystyle\begin{split}&S=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}\frac{\left(\sqrt{5}-1\right)}{2}&1&1&\frac{\left(\sqrt{5}+1\right)}{2}\\\
1&\frac{\left(1-\sqrt{5}\right)}{2}&\frac{\left(\sqrt{5}+1\right)}{2}&-1\\\
1&\frac{\left(\sqrt{5}+1\right)}{2}&\frac{\left(1-\sqrt{5}\right)}{2}&-1\\\
\frac{\left(\sqrt{5}+1\right)}{2}&-1&-1&\frac{\left(\sqrt{5}-1\right)}{2}\\\
\end{array}\right)\;\;,\\\
&\\{h_{\alpha}\\}_{\alpha=0}^{3}=\pm\left\\{0,\tfrac{3}{5},\tfrac{2}{5},0\right\\}\;.\end{split}$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,25,225,2500,30625,\ldots\right\\}\;.$

The modular structure is identical to that of $4_{0}^{B,c}$.

#### $S^{2}\left(2,5,\frac{5}{2}\right)$

There are four irreducible flat connections $\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{5}{2},\tfrac{5\sqrt{5}+3}{4},\tfrac{5(3-\sqrt{5})}{4},\tfrac{5}{2}\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{37}{40},\tfrac{29}{40},\tfrac{21}{40},\tfrac{13}{40}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is non-unitary. The
flat-connection-to-loop operator map is

$\displaystyle\rho_{\alpha=1}\rightarrow\left(a=x_{3}^{2},R=\Box\right)\;,\quad\rho_{\alpha=2}\rightarrow\left(a=x_{2},R=\Box\right)\;,$
$\displaystyle\rho_{\alpha=3}\rightarrow\left(a=x_{3}^{2},R=\Box\right)\otimes\left(a=x_{2},R=\Box\right)\;.$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{3}$ are

$\displaystyle
S=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}1&\frac{\left(1-\sqrt{5}\right)}{2}&-\frac{\left(1+\sqrt{5}\right)}{2}&1\\\
\frac{\left(1-\sqrt{5}\right)}{2}&-1&1&\frac{\left(\sqrt{5}+1\right)}{2}\\\
-\frac{\left(1+\sqrt{5}\right)}{2}&1&-1&\frac{\left(\sqrt{5}-1\right)}{2}\\\
1&\frac{\left(\sqrt{5}+1\right)}{2}&\frac{\left(\sqrt{5}-1\right)}{2}&1\\\
\end{array}\right)\;\;,$ (165)
$\displaystyle\\{h_{\alpha}\\}_{\alpha=0}^{3}=\textrm{diag}\left\\{0,\tfrac{4}{5},\tfrac{3}{5},\tfrac{2}{5}\right\\}\;.$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,25,225,2500,30625,\ldots\right\\}\;.$

The modular structure is identical to that of
$\left(\textrm{Gal}_{\sharp=2}[(A_{1},3)_{1/2}]\right)\otimes(A_{1},3)_{1/2}$.

#### $S^{2}\left(2,\frac{5}{2},\frac{5}{2}\right)$

There are four irreducible flat connections $\rho_{\alpha=0,1,2,3}$ with

$\displaystyle\begin{split}&\\{\textrm{Tor}[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{5(3-\sqrt{5})}{4},\tfrac{5}{2},\tfrac{5}{2},\tfrac{5(3+\sqrt{5})}{4}\right\\},\\\
&\\{CS[\rho_{\alpha}]\\}_{\alpha=0}^{3}=\left\\{\tfrac{3}{40},\tfrac{7}{8},\tfrac{7}{8},\tfrac{27}{40}\right\\}\;.\end{split}$

According to (21), the corresponding topological phase is non-unitary. The
flat-connection-to-loop operator map is

$\displaystyle\rho_{\alpha=1}\rightarrow\left(a=x_{3}^{2},R=\Box\right)\;,\quad\rho_{\alpha=2}\rightarrow\left(a=x_{2}^{2},R=\Box\right)\;,$
$\displaystyle\rho_{\alpha=3}\rightarrow\left(a=x_{3}^{2},R=\Box\right)\otimes\left(a=x_{2}^{2},R=\Box\right)\;.$

The $S$-matrix and the topological spins $\\{h_{\alpha}\\}_{\alpha=0}^{3}$ are

$\displaystyle
S=\frac{1}{\sqrt{5}}\left(\begin{array}[]{cccc}\frac{\left(\sqrt{5}+1\right)}{2}&-1&-1&\frac{\left(\sqrt{5}-1\right)}{2}\\\
-1&\frac{\left(\sqrt{5}-1\right)}{2}&\frac{\left(-\sqrt{5}-1\right)}{2}&1\\\
-1&\frac{\left(-\sqrt{5}-1\right)}{2}&\frac{\left(\sqrt{5}-1\right)}{2}&1\\\
\frac{\left(\sqrt{5}-1\right)}{2}&1&1&\frac{\left(\sqrt{5}+1\right)}{2}\\\
\end{array}\right)\;\;,$ (170)
$\displaystyle\\{h_{\alpha}\\}_{\alpha=0}^{3}=\pm\left\\{0,\tfrac{4}{5},\tfrac{4}{5},\tfrac{3}{5}\right\\}\;.$

The ground state degeneracy is

$\displaystyle\left\\{\textrm{GSD}_{g}\right\\}_{g=0,1,\ldots}=\left\\{1,4,25,225,2500,30625,\ldots\right\\}\;.$

The modular structure is identical to that of
$\left(\textrm{Gal}_{\sharp=2}[(A_{1},3)_{1/2}]\right)\otimes\left(\textrm{Gal}_{\sharp=2}[(A_{1},3)_{1/2}]\right)$.

###  F.2 Non-trivial $H_{1}(M,\mathbb{Z}_{2})$

#### $S^{2}(2,3,6)$

There are two irreducible $PSL(2,\mathbb{C})$ connections,
$\\{\rho^{PSL}_{\alpha}\\}_{\alpha=0}^{1}$, with trivial 2nd Stiefel-Whitney
class. There are two $\mathbb{Z}_{2}$ flat connections $\\{1,\eta\\}\in
H^{1}(M,\mathbb{Z}_{2})$. Among the two $PSL(2,\mathbb{C})$ connections, the
$\rho^{PSL}_{\alpha=1}$ is invariant under tensoring with $\eta$, i.e.

$\displaystyle\rho^{PSL}_{\alpha=1}\otimes\eta=\rho^{PSL}_{\alpha=1}\otimes
1\;\Rightarrow\;\textrm{Inv}(\rho_{\alpha=1})=\mathbb{Z}_{2}\;.$

Accoring to Table 5, there are $1+2*1=3$ anyons in TFT$[S^{2}(2,3,6)]$, which
are

$\displaystyle\rho^{PSL}_{\alpha=0}\;\textrm{ and
}\;\rho^{PSL}_{\alpha=1}\;(\textrm{with multiplicity 2})\;.$

We choose the flat connection which is not invariant under tensoring with
$\eta$ as the trivial anyon. Torsions for the flat connections are

$\displaystyle\textrm{Tor}[\rho_{\alpha=0}^{PSL}]=3\;,\quad\textrm{Tor}[\rho_{\alpha=1}^{PSL}]=\frac{3}{4}\;.$

From the dictionary in Table 5, we have

$\displaystyle(S_{0\alpha}^{2}\textrm{ for all three anyons})=\frac{1}{3}\;.$

There are 3 $SL(2,\mathbb{C})$ flat connections which are

$\displaystyle\rho^{PSL}_{\alpha=0}\otimes
1\;,\;\rho^{PSL}_{\alpha=0}\otimes\eta\;,\;\rho^{PSL}_{\alpha=1}\otimes 1\;.$

Since the $\rho^{PSL}_{\alpha=0,1}$ have trivial 2nd Stiefel-Whitney class,
$\rho^{PSL}_{\alpha=0,1}\otimes\delta$ with $\delta\in
H^{1}(M,\mathbb{Z}_{2})$ can be regarded as $SL(2,\mathbb{C})$ flat
connections on $M$. Their Chern-Simons actions are

$\displaystyle\begin{split}&CS[\rho_{\alpha=0}\otimes
1]=CS[\rho_{\alpha=0}\otimes\eta]=\frac{3}{4}\;,\\\ &CS[\rho_{\alpha=1}\otimes
1]=\frac{5}{12}\;.\end{split}$

From the 1st line, we can confirm that $\eta$ is bosonic and thus
TFT$[S^{2}(2,3,6)]$ is a bosonic topological theory, see (87) and (92). From
the computation of the Chern-Simons invariants, we have

$\displaystyle h(\rho^{PSL}_{\alpha=0})=0\;,\quad
h(\rho^{PSL}_{\alpha=1})=\pm\frac{1}{3}\;.$

Recall that there are two anyons associated to $\rho^{PSL}_{\alpha=1}$. The
spectrum of $\\{S_{0\alpha}^{2}\\}$ and $\\{h_{\alpha}\\}$ are identical to
the that of $3^{B}_{\pm 2}$.

This also implies that the mother theory
$\widetilde{\textrm{TFT}}[M=S^{2}(2,3,6)]$ is the $SU(2)_{4}$ theory. This
theory is a UMTC with 5 anyons including an anyon generating the 1-form
symmetry $H_{1}(M,\mathbb{Z}_{2})$. By condensing this symmetry generating
anyon, this theory reduces to the TFT$[S^{2}(2,3,6)]$.

#### $S^{2}(\frac{4}{q_{1}},\frac{4}{q_{2}},\frac{3}{2})$

There are three irreducible $PSL(2,\mathbb{C})$ connections,
$\\{\rho^{PSL}_{\alpha}\\}_{\alpha=0}^{2}$, with trivial 2nd Stiefel-Whitney
class. There are two $\mathbb{Z}_{2}$ flat connections $\\{1,\eta\\}\in
H^{1}(M,\mathbb{Z}_{2})$. Among the three $PSL(2,\mathbb{C})$ connections, the
$\rho^{PSL}_{\alpha=2}$ is invariant under tensoring with $\eta$, i.e.

$\displaystyle\rho^{PSL}_{\alpha=2}\otimes\eta=\rho^{PSL}_{\alpha=2}\otimes
1\;\Rightarrow\;\textrm{Inv}(\rho_{\alpha=2})=\mathbb{Z}_{2}\;.$

According to Table 5, there are $2+2*1=4$ anyons in TFT$[S^{2}(2,3,6)]$, which
are

$\displaystyle\rho^{PSL}_{\alpha=0}\;,\;\rho^{PSL}_{\alpha=1}\;\textrm{ and
}\;\rho^{PSL}_{\alpha=2}\;(\textrm{with multiplicity 2})\;.$

Torsions for the flat connections are

$\displaystyle\textrm{Tor}[\rho_{\alpha=0}^{PSL}]=\textrm{Tor}[\rho_{\alpha=1}^{PSL}]=4\;,\;\;\textrm{Tor}[\rho_{\alpha=2}^{PSL}]=1\;.$

From the dictionary in Table 5, we have

$\displaystyle(S_{0\alpha}^{2}\textrm{ for all four anyons})=\frac{1}{4}\;.$

There are 5 $SL(2,\mathbb{C})$ flat connections which are

$\displaystyle\rho^{PSL}_{\alpha=0,1}\otimes
1\;,\;\rho^{PSL}_{\alpha=0,1}\otimes\eta\;,\;\rho^{PSL}_{\alpha=2}\otimes
1\;.$

Since the $\rho^{PSL}_{\alpha=0,1}$ have trivial 2nd Stiefel-Whitney class,
$\rho^{PSL}_{\alpha=0,1}\otimes\delta$ with $\delta\in
H^{1}(M,\mathbb{Z}_{2})$ can be regarded as $SL(2,\mathbb{C})$ flat
connections on $M$. Their Chern-Simons actions are

$\displaystyle\begin{split}&CS[\rho_{\alpha=0}\otimes
1]=CS[\rho_{\alpha=0}\otimes\eta]=\pm\frac{\ell}{48}\;,\\\
&CS[\rho_{\alpha=1}\otimes
1]=CS[\rho_{\alpha=1}\otimes\eta]=\pm\left(\frac{\ell}{48}+\frac{1}{2}\right)\;,\\\
&CS[\rho_{\alpha=2}\otimes
1]=\pm\left(\frac{\ell}{48}+\frac{(\frac{q_{1}+q_{2}}{2}\textrm{ mod
4})}{8}\right)\;.\end{split}$

Here $\ell\in 2\mathbb{Z}$ depends on $\\{q_{i}\\}_{i=1}^{3}$. From the 1st
and 2nd line, we can confirm that $\eta$ is bosonic and thus
TFT$[S^{2}(\frac{4}{q_{1}},\frac{4}{q_{2}},\frac{3}{2})]$ is a bosonic
topological theory, see (87) and (92). From the computation of the Chern-
Simons invariants, we have

$\displaystyle\begin{split}&h(\rho^{PSL}_{\alpha=0})=0\;,\;h(\rho^{PSL}_{\alpha=2})=\pm\frac{1}{2}\;,\\\
&h(\rho^{PSL}_{\alpha=2})=\pm\frac{(\frac{q_{1}+q_{2}}{2}\textrm{ mod
4})}{8}\;.\end{split}$

Recall that there are two anyons associated to $\rho^{PSL}_{\alpha=2}$. The
spectrum of $\\{S_{0\alpha}^{2}\\}$ and $\\{h_{\alpha}\\}$ are identical to
the that of $4_{\pm(\frac{q_{1}+q_{2}}{2}\textrm{ mod 4})}^{B}$.

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$H_{1}(\partial W)\to H_{1}(W)$ restricted to $U$ is an isomorphism). Let $S$
be a genus 2 surface, which we will use to mark the boundaries of $W^{+}$ and
$W^{-}$. Assume $H_{1}(S;{\mathbb Z})$ is generated by
$e_{1},~{}e_{2},~{}e_{3},~{}e_{4}$. Choose a marking $S\to\partial W^{+}$ so
in $W^{+}$ one has $U=\langle e_{1},e_{2}\rangle$ and $V=\langle
e_{3},e_{4}\rangle$. Similarly, choose a marking $S\to\partial W^{-}$ so that
in $W^{-}$ one has $V=\langle e_{1},e_{2}\rangle$ and $U=\langle
e_{3},e_{4}\rangle$. We then define

$W_{n}=W^{+}\cup_{f^{n}}W^{-}$

where $f:S\to S$ is a pseudo-Anosov that acts on $H_{1}(S)$ by the symplectic
matrix

$F=\begin{pmatrix}2&1&0&0\\\ 1&1&0&0\\\ 0&0&1&-1\\\ 0&0&-1&2\end{pmatrix}$

For the existence of such a pseudoAnosov mapping class, see the proof Lemma
7.1 in [BD17]. This matrix preserves the subspace decomposition above, and so
ensures that every curve in $\partial W^{\pm}$ that is not nullhomologous in
$\partial W^{\pm}$ but bounds a surface in $M^{\pm}$ cannot bound on both
sides.

The mapping class $f$ acts as an Anosov matrix on $U$ and $V$. This ensures
the standard Euclidean $\ell^{2}$-norm $||F^{n}(a)||_{E}$ of an element $a\in
H_{1}(S)$ grows exponentially in $n$ (indeed, for our choice of $F$, it grows
like $(\frac{3+\sqrt{5}}{2})^{n}$). Since norms on finite dimensional real
vector spaces are comparable, there is a constant comparing the stable norm
induced by the metric inherited from $W$ to the standard Euclidean
$\ell^{2}$-norm on $H_{1}(S)$.

Lemma 7.3 in [BD17] explains how Theorem 8.1 in [BMNS16] implies that for
large $n$ the manifolds $W_{n}$ admit a $K$-biLipschitz diffeomorphism $\mu$
from the model manifold $M_{n}$ as described above. After increasing $K$, we
can drop the large $n$ condition. This then also implies the linear volume
growth and injectivity radius bounds. ∎

######  Remark 4.

Using the model manifold, one can easily estimate the Cheeger constant of
$W_{n}$, which will decay like $1/n$.

######  Theorem C.

The family $W_{n}$ of closed hyperbolic 3-manifolds from Proposition 6.3 has
1-form Laplacian spectral gap that vanishes exponentially fast in relation to
volume:

$\sqrt{\lambda(W_{n})}\leq
B\operatorname{vol}(W_{n})e^{-r\operatorname{vol}(W_{n})},$

where $r$ and $B$ are positive positive constants and $\lambda(W_{n})$ is the
first positive eigenvalue of the 1-form Laplacian on $W_{n}$.

###### Proof.

We continue using notation introduced in the previous propositions. Take
$\gamma$ in $\partial M^{+}\subset M_{n}$ to be an embedded geodesic loop
representing the class $e_{1}\in U\subset H_{1}(\partial W^{+})$. Recall from
the proof of Proposition 6.3 that $\gamma\subset\partial M^{+}$ does not bound
a surface in $M^{+}$ but that $f^{n}(\gamma)\subset\partial M^{-}$ bounds a
surface in $M^{-}$. Let $\alpha_{n}=f^{n}(\gamma)\subset\partial M^{-}\subset
M_{n}$. Note that $\alpha_{n}$ and $\gamma$ are isotopic in $M_{n}$.

Fix $\delta>0$. Consider some positive integer $m$ and incompressible surface
$\Sigma_{m}$ bounding $\alpha_{n}^{m}$ in $M_{n}$ with
$\chi_{-}(\Sigma_{m})/2m-\operatorname{{\tt{scl}}}_{M_{n}}(\gamma)<\delta$ and
which minimizes $\chi_{-}$ among surfaces with boundary $\alpha_{n}^{m}$. We
can then replace $\Sigma_{m}$ with a homotopic surface that pushes the
boundary of $\Sigma_{m}$ into the interior of $M^{-}$ and which intersects
$\partial M^{-}$ transversely and essentially in both $\partial M^{-}$ and
$\Sigma_{m}$. We can then attach an annulus to $\Sigma_{m}$ cobounding
$\alpha_{n}^{m}$ and the boundary of the modified surface $\Sigma_{m}$. This
new $\Sigma_{m}$ bounds $\alpha_{n}^{m}$ with a collar neighborhood of the
boundary contained entirely in $M^{-}$ and intersects $\partial M^{-}$
transversely in a union of loops essential in both $\Sigma_{m}$ and $\partial
M^{-}$.

We focus on the portion of $\Sigma_{m}$ that lies in $M^{-}$. Define
$\Sigma^{-}_{m}=\Sigma_{m}\cap M^{-}$. If $\Sigma_{m}$ is contained in
$M^{-}$, then Proposition 6.2 applied to $\alpha_{n}^{m}$ in $M^{-}$ implies
that

$||[\alpha_{n}^{m}]||_{s,\partial M^{-}}=m||[\alpha_{n}]||_{s,\partial
M^{-}}\leq m\operatorname{{\tt{scl}}}_{M^{-}}(\alpha_{n})\leq
D\chi_{-}(\Sigma_{m}^{-})=D\chi_{-}(\Sigma_{m}),$

where $||\cdot||_{s,\partial M^{-}}$ is the stable norm of $H_{1}(\partial
M^{-})$. Since
$\chi_{-}(\Sigma_{m})/m-\operatorname{{\tt{scl}}}_{M_{n}}(\gamma)\leq\delta$,
we conclude

$||[\alpha_{n}]||_{s,\partial M^{-}}\leq
D\operatorname{{\tt{scl}}}_{M_{n}}(\gamma)+D\delta.$

Our goal now is to get this same estimate for the other possible ways
$\Sigma_{m}$ sits in $M_{n}$.

2pt $M^{+}$ [ ] at 900 1560 $S\times[0,n]$ [ ] at 1480 1650 $M^{-}$ [ ] at
2000 1560 $\alpha$ [ ] at 1750 1050

Figure 4:  A schematic picture of the simplest case of a surface bounding
$\alpha$ in $M^{-}$.

Consider the case that $\Sigma_{m}$ does not lie entirely in $M^{-}$. There
are two possibilities. The first involves the surface $\Sigma_{m}$ passing
into the product region but not intersecting $M^{+}$. In this case the surface
can be homotoped to lie in $M^{-}$, so that Proposition 6.2 applies, giving
the desired estimate as in the previous case.

2pt $M^{+}$ [ ] at 900 1560 $S\times[0,n]$ [ ] at 1480 1650 $M^{-}$ [ ] at
2000 1560 $\alpha$ [ ] at 1750 850

Figure 5:  A schematic picture of a surface bounding $\alpha$ that passes back
into $M^{-}$ but does not pass into $M^{+}$.

2pt $M^{+}$ [ ] at 900 1560 $S\times[0,n]$ [ ] at 1480 1650 $M^{-}$ [ ] at
2000 1560 $\alpha$ [ ] at 1750 850 $c_{1}$ [ ] at 1750 1285 $c_{0}$ [ ] at
1750 1070

Figure 6:  A schematic picture of a surface bounding $\alpha$ that passes back
into $M^{+}$. Notice the multicurve $c^{-}=c_{0}\cup c_{1}$ bounds surfaces in
$M^{+}$ and $M^{-}$, so is homologically trivial.

The second possibility concerns the surface $\Sigma_{m}$ crossing through the
product region into $M^{+}$ with an essential intersection with
$\partial\Sigma^{+}$. In this case, we will see that the surface
$\Sigma_{m}^{-}$ has boundary homologous to $\alpha_{n}^{m}$, which will allow
us to apply Proposition 6.2 to obtain the desired estimate. By construction, a
sufficiently small collar $C$ of the boundary $\partial\Sigma_{m}$ in
$\Sigma_{m}$ maps into $M^{-}$, so in particular, a subsurface of
$\Sigma_{m}^{-}$ has some boundary component that maps to $\alpha_{n}^{m}$.
That boundary component can be closed by attaching a surface $S^{-}$ that
bounds $\alpha_{n}^{m}$ in $M^{-}$ to $\Sigma_{m}$. From this, we see that the
multicurve $c^{-}=\partial\Sigma^{-}_{m}-\alpha_{n}$ bounds surfaces in
$M^{+}$ and $M^{-}$. Thus by Proposition 6.3, $c^{-}$ must be homologically
trivial in $\partial M^{-}$. Let
$x=\partial\Sigma_{m}^{-}=c^{-}+\alpha_{n}^{m}$. Since $c^{-}$ is
nullhomologous, $||[x]||_{s,\partial M^{-}}=||[\alpha_{n}^{m}]||_{s,\partial
M^{-}}=m||[\alpha_{n}]||_{s,\partial M^{-}}.$ By Proposition 6.2 ,
$||[x]||_{s,\partial M^{-}}\leq D\operatorname{{\tt{scl}}}_{M^{-}}(x).$ Since
$x$ is essential in $\Sigma_{m}$, we get that
$\chi_{-}(\Sigma_{m}^{-})\leq\chi_{-}(\Sigma_{m})$, then using that
$\chi_{-}(\Sigma_{m})/2m-\delta\leq\operatorname{{\tt{scl}}}_{M_{n}}(\alpha_{n})$,
we obtain
$\operatorname{{\tt{scl}}}_{M^{-}}(x)\leq\chi_{-}(\Sigma_{m}^{-})/2\leq
m\operatorname{{\tt{scl}}}_{M_{n}}(\alpha_{n})+\delta m$. Putting this all
together and dividing by $m$, we get that

$||[\alpha_{n}]||_{s,\partial M^{-}}\leq
D\operatorname{{\tt{scl}}}_{M_{n}}(\alpha_{n})+D\delta.$

We therefore have in each case that there is a constant $D$ independent of $n$
such that

$||[\alpha_{n}]||_{s,\partial M^{-}}\leq
D\operatorname{{\tt{scl}}}_{M_{n}}(\alpha_{n})+D\delta.$

By Proposition 6.3 , $||[\alpha_{n}]||_{s,\partial
M^{-}}=||[f^{n}(\gamma)]||_{s,\partial M^{+}}$ grows exponentially in $n$.
Thus for some constants $B>0$ and $r>0$, we have

$Be^{rn}\leq D\operatorname{{\tt{scl}}}_{M_{n}}(\gamma)+D\delta,$

where we use that $\gamma$ and $\alpha_{n}$ are homotopic in $M_{n}$. Using
the injectivity radius lower bound and Theorem 3.9 of [Cal09], we can increase
$D$ and drop the additive constant in this inequality. By Proposition 6.3 ,
the volume growth of the $W_{n}$ is proportional to $n$ so there is a constant
$C$ such that $\operatorname{vol}(W_{n})\leq Cn$. Additionally, using the
$K$-biLipschitz comparison of Proposition 6.3 , the length of $\gamma$ in
$W_{n}$ is bounded from above by $2K|\gamma|_{W}$, where $W$ is the tripus. As
a result, Theorem A implies that the spectral gap for the 1-form Laplacian of
the manifolds $W_{n}$ vanishes exponentially fast in $n$. In particular, we
have

$\sqrt{\lambda(W_{n})}\leq
A\operatorname{vol}(W_{n})\frac{|\gamma|_{W_{n}}}{\operatorname{{\tt{scl}}}_{W_{n}}(\gamma)}\leq
2KACB^{-1}D|\gamma|_{W}ne^{-rn},$

so the result holds after redefining $B$ to be $2KACB^{-1}D|\gamma|_{W}.$

∎

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