Exercise 58.2

Question

For each of the following spaces, the fundamental group is either trivial, infinite cyclic, or isomorphic to the fundamental group of the figure-eight. Determine for each space which of the three alternatives holds.
\begin{enumerate}[label=(\alph*)]
\item The ``solid torus,'' $B^2 	imes S^1$.
\item The torus $T$ with a point removed.
\item The cylinder $S^1 	imes \mathbb{R}$.
\item The infinite cylinder $S^1 	imes \mathbb{R}$.
\item $\mathbb{R}^3$ with the nonnegative $x$, $y$, and $z$ axes deleted.
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The following subsets of $\mathbb{R}^2$:
\item $\{x \mid \Vert x \Vert > 1\}$
\item $\{x \mid \Vert x \Vert \ge 1\}$
\item $\{x \mid \Vert x \Vert < 1\}$
\item $S^1 \cup (\mathbb{R}_+ 	imes 0)$
\item $S^1 \cup (\mathbb{R}_+ 	imes \mathbb{R})$
\item $S^1 \cup (\mathbb{R} 	imes 0)$
\item $\mathbb{R}^2 - (\mathbb{R}_+ 	imes 0)$
\end{enumerate}

Amit Awasthi · Accepted Answer

\begin{proof}[Solution]
  $(a)$. $\pi_1(B^2 	imes S^1) = \pi_1(B^2) 	imes \pi_1(S^1) = \pi_1(S^1)$, for we have a deformation retraction from $B^2$ onto $\{0\}$.
  \par $(b)$. $\pi_1(T \setminus \{p\})$ is isomorphic to the fundamental group
  of the figure-eight, for we have a deformation retraction onto the
  figure-eight by deforming $T \setminus \{p\}$ in the following manner:
  \begin{center}
    \begin{tikzpicture}[line cap=round,line join=round,>=stealth,x=1.0cm,y=1.0cm]
      
ode (a) at (0,0) {
        \begin{tikzpicture}[>=stealth]
          \fill [gray] (0,0) rectangle (2,2);
          \draw (0,1) node[anchor=east] {$a$};
          \draw (2,1) node[anchor=west] {$a$};
          \draw (1,0) node[anchor=north] {$b$};
          \draw (1,2) node[anchor=south] {$b$};
          \draw [->-=0.55] (0,0) -- (0,2);
          \draw [->-=0.55] (0,2) -- (2,2);
          \draw [->-=0.55] (2,0) -- (2,2);
          \draw [->-=0.55] (0,0) -- (2,0);
          \fill [white] (1,1) circle (0.1); 
          \draw (1,1) circle(0.1);
        \end{tikzpicture}
      };
      
ode (b) at (a.east) [anchor=west,xshift=0.5cm] {
        \begin{tikzpicture}[>=stealth]
          \fill [gray] (0,0) rectangle (2,2);
          \draw (0,1) node[anchor=east] {$a$};
          \draw (2,1) node[anchor=west] {$a$};
          \draw (1,0) node[anchor=north] {$b$};
          \draw (1,2) node[anchor=south] {$b$};
          \draw [->-=0.55] (0,0) -- (0,2);
          \draw [->-=0.55] (0,2) -- (2,2);
          \draw [->-=0.55] (2,0) -- (2,2);
          \draw [->-=0.55] (0,0) -- (2,0);

\draw [->--=0.55] (1,1) -- (2,0);
          \draw [->--=0.55] (1,1) -- (2,2);
          \draw [->--=0.55] (1,1) -- (0,0);
          \draw [->--=0.55] (1,1) -- (0,2);
          \draw [->--=0.56] (1,1) -- (1,0);
          \draw [->--=0.56] (1,1) -- (1,2);
          \draw [->--=0.56] (1,1) -- (0,1);
          \draw [->--=0.56] (1,1) -- (2,1);

\fill [white] (1,1) circle (0.1); 
          \draw (1,1) circle(0.1);
        \end{tikzpicture}

};
      
ode (c) at (b.east) [anchor=west,xshift=0.5cm] {
        \begin{tikzpicture}[>=stealth]
          \draw (0,1) node[anchor=east] {$a$};
          \draw (2,1) node[anchor=west] {$a$};
          \draw (1,0) node[anchor=north] {$b$};
          \draw (1,2) node[anchor=south] {$b$};
          
          \draw [->-=0.55] (0,0) -- (0,2);
          \draw [->-=0.55] (0,2) -- (2,2);
          \draw [->-=0.55] (2,0) -- (2,2);
          \draw [->-=0.55] (0,0) -- (2,0);
        \end{tikzpicture}
      };

ode (d) at (c.east) [anchor=west,xshift=0.5cm]{
        \begin{tikzpicture}[>=stealth]
          \draw (-1,0) node[anchor=east] {$a$};
          \draw (1,0) node[anchor=west] {$b$};
          \fill (0,0) circle (0.066);
          \draw[postaction={decoration={markings,mark=at position 0.75 with {\arrow[scale=2]{>}}},decorate}] (-0.5,0) circle (0.5);
          \draw[postaction={decoration={markings,mark=at position 0.25 with {\arrow[scale=2]{>}}},decorate}] (0.5,0) circle (0.5);
        \end{tikzpicture}
      };
      \path[-to,line width=0.5pt] (a) edge (b);
      \path[-to,line width=0.5pt] (b) edge (c);
      \path[-to,line width=0.5pt] (c) edge (d);
    \end{tikzpicture}
  \end{center}
  where the last arrow comes from the construction of the torus as the quotient
  $\mathbb{R}^2/\mathbb{Z}^2$.
  \par $(c)$. $\pi_1(S^1 	imes I) = \pi_1(S^1) 	imes \pi_1(I) = \pi_1(S^1)$, for we have the deformation retraction from $I$ onto $\{0\}$.
  \par $(d)$. $\pi_1(S^1 	imes \mathbb{R}) = \pi_1(S^1) 	imes \pi_1(\mathbb{R})$, for we have the deformation retraction from $\mathbb{R}$ onto $\{0\}$.
  \par $(e)$. This space retracts onto $S^2 \setminus \{p,q,r\} \simeq \mathbb{R}^2 \setminus \{s,t\}$ (where the isomorphism is from Theorem $59.3$), which retracts onto the figure-eight, and so the fundamental group is isomorphic to the fundamental group of the figure-eight.
  \par $(f)$. $\pi_1(\{x \mid \Vert x \Vert > 1\}) = \pi_1(S^1)$, by a deformation retraction onto a circle of radius $>1$, whose fundamental group is $\pi_1(S^1)$ since it is homeomorphic to $S^1$.
  \par $(g)$. $\pi_1(\{x \mid \Vert x \Vert \ge 1\}) = \pi_s(S^1)$, for we have the deformation retraction onto $S^1$.
  \par $(h)$. $\pi_1(\{x \mid \Vert x \Vert < 1\}) = \pi_1(\{0\}) = 1$, by a deformation retraction onto $\{0\}$.
  \par $(i)$. $\pi_1(S^1 \cup (\mathbb{R}_+ 	imes 0)) = \pi_1(S^1)$, since we can retract $\mathbb{R}_+$ to one point $(1,0) \in S^1$.
  \par $(j)$. $\pi_1(S^1 \cup (\mathbb{R}_+ 	imes \mathbb{R})) = \pi_1(S^1)$, since we can retract $\mathbb{R}_+ 	imes \mathbb{R}$ to the half-circle $S^1 \cap \mathbb{R}_+ 	imes \mathbb{R}$.
  \par $(k)$. $\pi_1(S^1 \cup (\mathbb{R} 	imes 0))$ is isomorphic to the
  fundamental group of the figure-eight, for we can retract $(\mathbb{R} 	imes
  0)$ onto the line segment from $(-1,0)$ to $(1,0)$, which gives a topological
  space homotopy equivalent to the figure-eight.
  \par $(l)$. $\pi_1(\mathbb{R}^2 \setminus (\mathbb{R}_+ 	imes 0)) = 1$, since we can retract onto any arbitrary point $p \in \mathbb{R}^2 \setminus (\mathbb{R}_+ 	imes 0)$.
\end{proof}

Exercise 58.2

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Exercise 58.2

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