Exercises 1.5 Problem set 2, due 19 October 2021
1.
Let \(n\in\NN\text{.}\) Construct a homeomorphism \(f \colon \RR^n \to S^n\smallsetminus\{(1,0,0,\dots,0\}\text{.}\)
2.
Define subspaces
\begin{equation*}
D^2 = \{ (u,v) \in \RR^2 : \|(u,v)\| \leq 1\} \subset \RR^2
\end{equation*}
and
\begin{equation*}
ST = \{(x,y,z)\in \RR^3 : (2-\|(x,y)\|)^2 + z^2 \leq 1\} \subset \RR^3\text{.}
\end{equation*}
Construct a homeomorphism \(g \colon D^2 \times S^1 \to ST\text{.}\)
3.
Let us use the notation from the previous two problems. Consider the subspace
\begin{equation*}
B^2 = \{ (u,v) \in \RR^2 : \|(u,v)\| \lt 1\} \subset \RR^2\text{.}
\end{equation*}
Prove that \(S^3 \smallsetminus f(g(B^2 \times S^1))\) is homeomorphic to \(ST\text{.}\)