Exercise 73.1

Question

Find spaces whose fundamental groups are isomorphic to the following groups. (Here $\mathbb{Z}/n$ denotes the additive group of integers modulo $n$.)
\begin{enumerate}[label=(\alph*)]
\item $\mathbb{Z}/n 	imes \mathbb{Z}/m$.
\item $\mathbb{Z}/n_1 	imes \mathbb{Z}/n_2 	imes \cdots 	imes \mathbb{Z}/n_k$.
\item $\mathbb{Z}/n * \mathbb{Z}/m$. (See \hyperref[exc:71.2]{Exercise $2$ of
    $\S71$}).
\item $\mathbb{Z}/n_1 * \mathbb{Z}/n_2 * \cdots * \mathbb{Z}/n_k$.
\end{enumerate}

Amit Awasthi · Accepted Answer

$\DeclareMathOperator*{\bigast}{\ast}$
\begin{proof}[Solution]
  We use Theorem $60.1$ to say $\pi_1(X 	imes Y,x_0 	imes y_0) = \pi_1(X,x_0)
  	imes \pi_1(Y,y_0)$, and use \href{exercise_71-2}{Exercise 71.2} to say $\pi_1(X \vee Y,x_0) = \pi_1(X,x_0) * \pi_1(Y,x_0)$ for $\{x_0\} = X \cap Y$.
  \par We denote the $n$-fold dunce cap by $D_n$. Note since the $D_n$ are path connected, we do not need to specify base points.
  \par Through repeated applications of the above, and by using induction to get $(a) \Rightarrow (b), (c) \Rightarrow (d)$, we get
  \begin{alignat*}{4}
    \pi_1(D_n 	imes D_m) &= \mathbb{Z}/n 	imes \mathbb{Z}/m &\implies{}& \pi_1\left(\prod_{i=1}^k D_{n_i}ight) &={}& \prod_{i=1}^k \mathbb{Z}/n_i,\
    \pi_1(D_n \vee D_m) &= \mathbb{Z}/n * \mathbb{Z}/m &\implies{}& \pi_1\left( \bigvee_{i=1}^k D_{n_i} ight) &={}& \bigast_{i=1}^k \mathbb{Z}/n_i.
  \end{alignat*}
\end{proof}

Exercise 73.1

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Exercise 73.1

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