Exercises:Mond - Topology - 1/Question 7

Section B

Question 7

Let $D^2$ denote the closed unit disk in $\mathbb{R}^2$ and define an equivalence relation on $D^2$ by setting $x_1\sim x_2$ if $\Vert x_1\Vert=\Vert x_2\Vert=1$ ("collapsing the boundary to a single point"). Show that $\frac{D^2}{\sim}$ is homeomorphic to $\mathbb{S}^2$ - the sphere.

• Hint: first define a surjection $(:D^2\rightarrow\mathbb{S}^2)$ mapping all of $\partial D^2$ to the north pole. This may be defined using a good picture or a formula.

Solution

The idea is to double the radius of $D^2$, then pop it out into a hemisphere, then pull the rim to a point

Picture showing the "expanding $D^2$", the embedding-in-$\mathbb{R}^3$ part, and the "popping out"

Definitions:

• $H$ denotes the hemisphere in my picture.
• $E:D^2\rightarrow H$ is the composition of maps in my diagram that take $D^2$, double its radius, then embed it in $\mathbb{R}^3$ then "pop it out" into a hemisphere. We take it as obvious that it is a homeomorphism
• $f':H\rightarrow\mathbb{S}^2$, this is the map in the top picture. It takes the hemisphere and pulls the boundary/rim in (along the blue lines) to the north pole of the red sphere. $f'(\partial H)=(0,0,1)\in\mathbb{R}^3$, it should be clear that for all $x\in H-\partial H$ that $f'(x)$ is intended to be a point on the red sphere and that $f'\big\vert_{H-\partial H}$ is injective. It is also taken as clear that $f'$ is surjective
• Note: Click the pictures for a larger version
• $\frac{D^2}{\sim}$ and $D^2/\sim$ denote the quotient space, with this definition we get a canonical projection, $\pi:D^2\rightarrow D^2/\sim$ given by $\pi:x\mapsto [x]$ where $[x]$ denotes the equivalence class of $x$
• Lastly, we define $f:D^2\rightarrow\mathbb{S}^2$ to be the composition of $E$ and $f'$, that is: $f:=f'\circ E$, meaning $f:x\mapsto f'(E(x))$

The situation is shown diagramatically below:

• $\xymatrix{ D^2 \ar[d]_\pi \ar[r]^E \ar@/^1.5pc/[rr]^{f:=f'\circ E} & H \ar[r]^{f'} & \mathbb{S}^2 \\ \frac{D^2}{\sim} }$

Outline of the solution:

• We then want apply the passing to the quotient theorem to yield a commutative diagram: $\xymatrix{ D^2 \ar[d]_\pi \ar[r]^E \ar@/^1.5pc/[rr]^{f} & H \ar[r]^{f'} & \mathbb{S}^2 \\ \frac{D^2}{\sim} \ar@{.>}[urr]_{\bar{f} } }$
  • The commutative diagram part merely means that $f=\bar{f}\circ\pi$. We get $f=\bar{f}\circ\pi$ as a result of the passing-to-the-quotient theorem.
  • We take this diagram as showing morphisms in the TOP category, meaning all arrows shown represent continuous maps. (Obviously...)
• Lastly, we will show that $\bar{f}$ is a homeomorphism using the compact-to-Hausdorff theorem

Solution: