For the quotient space $G=G_0/G_1$, knowing the homotopy
groups of $G_0$ and $G_1$, one can determine homotopy groups from the long
exact sequence

$$
...
\to \pi_n(G_1)     \to \pi_n(G_0)     \to \pi_n(G_0/G_1)
\to \pi_{n-1}(G_1) \to \pi_{n-1}(G_0) \to \pi_{n-1}(G_0/G_1) \to ....
$$
But in practice, there are some subtleties.

Consider this particular group: 

>$${G}=\frac{SU(N)_A \times SU(N)_{B}  \times U(1)}{(\mathbb{Z}_N)^2} \equiv \frac{G_0}{G_1},$$

We define $G_0 \equiv SU(N)_A \times SU(N)_{B}  \times U(1)$ and $G_1 \equiv {(\mathbb{Z}_N)^2}$.

> **Question**: What is the fundamental group  $\pi_1(G)=\pi_1(G_0/G_1)=?$


----

This group can be understood as a triplet
$$(g_A, g_B, e^{i \theta}) \in SU(N)_A \times SU(N)_B \times U(1),$$
such that
$(e^{i \frac{2\pi}{N}} \mathbb{I}_N, \mathbb{I}_N, e^{-i \frac{2\pi}{N}})$ is the first $\mathbb{Z}_N$ generator mod out in $G$, while $(\mathbb{I}_N,e^{i \frac{2\pi}{N}} \mathbb{I}_N, e^{-i \frac{2\pi}{N}})$ is the second $\mathbb{Z}_N$ generator mod out in $G$. 

The $\mathbb{I}_N$ means the rank-N identity matrix.

Namely (1) the center of $SU(N)_A$, (2) the center of $SU(N)_B$ and (3) the $e^{i \frac{2\pi}{N}}\in U(1)$ overlap, thus we only mod out the twice redundant $(\mathbb{Z}_N)^2$. 

In other words, the following three $\mathbb{Z}_N$ generators within $G_0$ are identified:
$$
(e^{i \frac{2\pi}{N}} \mathbb{I}_N, \mathbb{I}_N, 1)
\simeq
( \mathbb{I}_N, e^{i \frac{2\pi}{N}} \mathbb{I}_N, 1)
\simeq
( \mathbb{I}_N, \mathbb{I}_N, e^{i \frac{2\pi}{N}}),
$$
that is why we mod out $G_1={(\mathbb{Z}_N)^2}$ out of $G_0$, when we define ${G} \equiv \frac{G_0}{G_1}$ earlier.

**Attempt**:

One can see that
$$
...
\to \pi_1(G_1) =0    \to \pi_1(G_0) = \mathbb{Z}   \to \pi_1(G_0/G_1)$$
$$\to \pi_{0}(G_1)=(\mathbb{Z}_N)^2 \to \pi_{0}(G_0)=0 \to \pi_{0}(G_0/G_1)=0 
$$
so to speak,
$$
0    \to \mathbb{Z}   \to \pi_1(G_0/G_1)\to (\mathbb{Z}_N)^2 \to  0 \to 0,
\tag{i} 
$$
so there is a short exact sequence determine the fundamental group $\pi_1(G_0/G_1)$, this is  possible a non-Abelian group? And this extension can be classified by the cohomology group $H^2(B(\mathbb{Z}_N)^2, \mathbb{Z}  )=H^1(B(\mathbb{Z}_N)^2, U(1)  )=(\mathbb{Z}_N)^2$?

**Attempt 2**: By the earlier definition, one can rewrite 

>$${G}=\frac{SU(N)_A \times SU(N)_{B}  \times U(1)}{(\mathbb{Z}_N)^2}
= \frac{U(N)_A \times SU(N)_{B}}{(\mathbb{Z}_N)}
\equiv \frac{G_0'}{G_1'},$$

by the fact that $U(N)_A \equiv \frac{SU(N)_A\times U(1)}{(\mathbb{Z}_N)}$,
$G_0' \equiv U(N)_A \times SU(N)_{B}$, $G_1' \equiv \mathbb{Z}_N$.
Now compute again 
$$
...
\to \pi_1(G_1') =0    \to \pi_1(G_0') = \mathbb{Z}   \to \pi_1(G_1'/G_0') $$
$$\to \pi_{0}(G_1')=\mathbb{Z}_N \to \pi_{0}(G_0')=0 \to \pi_{0}(G)=0 
$$
so to speak,
$$
 0    \to \mathbb{Z}   \to \pi_1(G)=\pi_1(G_1'/G_0')\to \mathbb{Z}_N \to  0 \to 0. \tag{ii}
$$
And this distinct extension can be classified by a distinct cohomology group $H^2(B\mathbb{Z}_N, \mathbb{Z}  )=H^1(B\mathbb{Z}_N, U(1)  )=\mathbb{Z}_N$?


Since 
>$\pi_1(G)=\pi_1(G_1/G_0)=\pi_1(G_1'/G_0')$, 

so the answers from **Attempt 1** and **Attempt 2** must be the same, from two different short exact sequences (i) and (ii). What is the unique answer?

>Is $\pi_1(G)=\mathbb{Z}$ or $(\mathbb{Z}  \rtimes \mathbb{Z}_N)$, or something else? We can take $N=3$ as a specific example.