Exercise 75.3

Question

Let $X$ be the quotient space obtained from an $8$-sided polygonal region $P$ by pasting its edges together according to the labelling scheme $acadbcb^{-1}d$.
\begin{enumerate}[label=(\alph*)]
\item Check that all vertices of $P$ are mapped to the same point of the quotient space $X$ by the pasting map.
\item Calculate $H_1(X)$.
\item Assuming $X$ is homeomorphic to one of the surfaces given in Theorem $75.5$ (which it is), which surface is it?
\end{enumerate}

Amit Awasthi · Accepted Answer

\begin{figure}[h]
  \centering
  \begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1.0cm,y=1.0cm]
    \draw [->,dotted] (0,0) -- (0,1);
    \draw [->,dotted] (0,1) -- (0.71,1.71);
    \draw [->,dotted] (0.71,1.71) -- (1.71,1.71);
    \draw [->,dotted] (1.71,1.71) -- (2.41,1);
    \draw [->,dotted] (2.41,1) -- (2.41,0);
    \draw [->,dotted] (2.41,0) -- (1.71,-0.71);
    \draw [->,dotted] (0.71,-0.71) -- (1.71,-0.71);
    \draw [->,dotted] (0.71,-0.71) -- (0,0);
    \draw [-] (0,1) -- (1.71,1.71);
    \draw [-] (1.71,1.71) -- (0.71,-0.71);
    \draw [-] (0.71,-0.71) -- (2.41,1);
    \draw [-] (2.41,1) -- (0,0);
    \draw [-] (0,0) -- (0.71,1.71);
    \draw [-] (0.71,1.71) -- (1.71,-0.71);
    \draw [-] (0,1) -- (2.41,0);
    \draw [-] (2.41,0) -- (1.71,-0.71);
    \draw (-0.46,0.7) node[anchor=north west] {$a$};
    \draw (0.89,2.15) node[anchor=north west] {$a$};
    \draw (-0.04,1.7) node[anchor=north west] {$c$};
    \draw (1.94,1.92) node[anchor=north west] {$d$};
    \draw (-0.05,-0.27) node[anchor=north west] {$d$};
    \draw (2.35,0.96) node[anchor=north west] {$b$};
    \draw (0.82,-0.68) node[anchor=north west] {$b^{-1}$};
    \draw (2.05,-0.15) node[anchor=north west] {$c$};
  \end{tikzpicture}
  \caption{Labeling of edges and identification of vertices in $P$.}
  \label{fig5}
\end{figure}
\begin{proof}[Proof of $(a)$]
  We have the identification of vertices in $P$ as shown with solid lines in Figure \href{fig5}, found by identifying heads/tails of arrows with the same label. Thus, all the vertices of $P$ are mapped to the same point of the quotient space $X$ by the pasting map.
\end{proof}
\begin{proof}[Solution for $(b)$]
  We apply Lemma $77.1$ to the labeling scheme $acadbcb^{-1}d$ repeatedly to find an equivalent labeling scheme, where the brackets show our decomposition $[y_0]a[y_1]a[y_2]$:
  \begin{align*}
    [~]a[c]a[dbcb^{-1}d] &\sim aac^{-1}dbcb^{-1}d\
    [aac^{-1}]d[bcb^{-1}]d[~] &\sim ddaac^{-1}bc^{-1}b^{-1}\
    [ddaa]c^{-1}[b]c^{-1}[b^{-1}] &\sim c^{-1}c^{-1}ddaab^{-1}b^{-1}.
  \end{align*}
  Relabeling the edges, which is allowed by p.~460, we have the labeling scheme $aabbccdd$. Thus, by Theorem $74.2$, the fundamental group is
  \begin{equation*}
    \pi_1(X,x_0) = \braket{a,b,c,d \mid a^2b^2c^2d^2 = 1}.
  \end{equation*}
  Finally, the first homology group is
  \begin{equation*}
    H_1(X) \simeq \braket{a,b,c,d \mid 2a + 2b + 2c + 2d = 0} \simeq
    \mathbb{Z}^3 	imes (\mathbb{Z}/2\mathbb{Z}).
  \end{equation*}
\end{proof}
\begin{proof}[Solution for $(c)$]
  Theorem $75.4$ says that $H_1(P_n) = \mathbb{Z}^{n-1} 	imes (\mathbb{Z}/2\mathbb{Z})$. Thus, assuming that $X$ is homeomorphic to one of the surfaces in Theorem $75.5$, we see that $X$ is homeomorphic to $P_4 = P^2\#P^2\#P^2\#P^2$ by $(b)$.
\end{proof}

Exercise 75.3

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Exercise 75.3

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