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Constructions

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Gluing things together

We can build new things from old by gluing:

Gluing diagram.png

In fact, we can build a lot of topologically different things with nothing but sheets of paper and a glue-stick:

Glue edges of paper.png

What is behind this gluing metaphor is an equivalence relation. Its axioms will then make practical sense. The Reflexivity Axiom, $A \sim A$, is: every spot of the sheet is glued to itself. The Symmetry Axiom, $A \sim B \Longrightarrow B \sim A$, becomes: a drop of glue holds either of the two sheets equally well.

Most of the time, we will attach the sheets edge-to-edge, without overlap. In that case, welding may be a better metaphor:

Welding.png

Then the Transitivity Axiom, $A \sim B, B \sim C \Longrightarrow A \sim C$, means: the two seams fully merge and become indistinguishable.

Thinking of a zipper is also appropriate:

Zipping jacket.png

Topologically, zipping a jacket turns a square with two holes into a cylinder with two holes, or, even better, a disk with two holes turns into a disk with three holes:

Zipping jacket 2.png

Making balloon animals is another example:

Balloon animals.png

Meanwhile, we have already seen gluing when we constructed topological spaces from cells as realizations of simplicial complexes:

These realizations, however, were placed within a specific Euclidean space ${\bf R}^N$. We will see that this is unnecessary.

Quotient sets

Before we consider the topological issues, let's make clear what happens to the underlying sets first.

We pick two examples of somewhat different nature.

Example (circle). The easiest way to construct the circle ${\bf S}^1$ is to take the closed segment $[0,1]$ and glue the endpoints together. The gluing procedure is expressed by the following equivalence relation on $X = [0,1]$:

  • 1. $0 \sim 1$, and
  • 2. $x \sim x$ for all $x\in X$. $\\$

The second condition is required by the Reflexivity Axiom and will be assumed implicitly in all examples. Then we record the equivalence relation simply as $$0 \sim 1.$$ We use the following notation for the quotient set: $$X / _{\sim} := \{[x]: x\in X\}.$$ As we know, it is simply the set of all equivalence classes of this equivalence relation: $${\bf S}^1 := [0,1]/_{\sim}.$$ where $$[x]: = \{y\in X: y \sim x\}= \begin{cases} \{0,1\} &\text{ if } x=0,1; \\ \{x\} &\text{ if } x\in (0,1). \end{cases}$$ The image below is just an illustration of what these equivalence classes look like:

Quotient - circle from segment.png

Only one of them is non-trivial. $\square$

Example (plane). The next example is a familiar one from linear algebra. We choose

  • $X = {\bf R}^2$ and
  • $(x,y) \sim (x,y')$ for all $x,y,y'\in {\bf R}$.
Equivalence relation on R2.png

The equivalence classes are the vertical lines. Instead of using the gluing metaphor, we say that each of these lines collapse to a point. The reason is that each of the vertical lines corresponds to its point of intersection with the $x$-axis. Hence, the quotient set corresponds, in this sense, to the real line. Algebraically, we need to find a map: $$\hspace{.18in}q:X/_{\sim} = \big \{\{(x,y): y\in {\bf R}\} :\ x\in {\bf R}\big \} \to \{x :\ x\in {\bf R}\} = {\bf R}.\hspace{.18in}\square$$

Definition. The function that takes each point to its equivalence class is called the identification function:

  • $q: X \to X/_{\sim}$ given by
  • $q(x) := [x]$. $\\$

It may be called the “gluing”, or “attaching”, map in a topological context.

When we recognize the quotient as a familiar set, we may try to present the identification map explicitly. In the first example, the identification function $$q: [0,1] \to {\bf S}^1$$ is given by $$q(t) := (\cos(\pi t), \sin(\pi t)),\ t\in [0,1].$$ In the second example, $q$ may be thought of as the projection: $$q: {\bf R}^2 \to {\bf R},$$ given by $$q(x,y) := x,\ x,y\in {\bf R}.$$

Two harder examples of gluing follow.

Example (surfaces). One can glue the two opposite edges of the square to create a cylinder:

Cylinder construction.png

If you twist the edge before gluing, you get the Möbius band:

Mobius band construction.png

Now, in order to properly interpret this literal gluing in terms of quotients, we need to recognize that, mathematically, only points can be identified to each other. In other words, points are glued pairwise:

Square glue to cylinder.png

Once this is understood, we can sometimes see a pattern of how the points are glued to each other and think of this process as gluing two whole edges, as long as they are oriented properly! These orientations of the edges are shown with arrows that have to be aligned before being glued.

We have here: $$X = [0,1] \times [0,1] = \{(x,y):\ x\in [0,1], y\in [0,1]\}.$$ And the equivalence relation for the cylinder is given by:

  • $(x,y) \sim (u,v)$ if $y=v$ and $x,u=0$ or $1$, $\\$

or simply:

  • $(0,y) \sim (1,y)$. $\\$

The equivalence relation for the Möbius band is given by:

  • $(x,y) \sim (u,v)$ if $y=1-v$ and $x,u=0$ or $1$, $\\$

or simply:

  • $(0,y) \sim (1,1-y)$. $\square$

Identification maps

Example. Note that the identification functions in the examples above are familiar. In the second example, $q$ is the projection: $$q : {\bf R}^2 \to {\bf R}$$ given by $$q(x,y) := x.$$

In the first example, the identification function $f$ is the gluing map: $$q: [0,1] \to {\bf S}^1,$$ given by $$q(t) := ( \cos(\pi t), \sin(\pi t) ), \ \forall t \in [0,1].$$ $\square$

Example (plane). The equivalence classes are the vertical lines and, therefore, the preimage of any subset $U \subset X/_{\sim}$ is made of vertical lines.

Equivalence relation on R2 identification map.png

$\square$

Example. Let's choose $f=q$ in the two examples of maps from the last subsection, $$f:[0,1]\to {\bf S}^1$$ and $$f:{\bf R} \times {\bf R} \to {\bf R}.$$ The idea is illustrated below:

Maps as identifications.png

$\square$

Example (folding). Fold the $xy$-plane, $$f(x,y) := (|x|,y).$$

Fold xy plane.png

Now, the folding of the rectangle $[-1,1] \times [0,1]$ is given by the same formula and is continuous as a restriction of $f$.

Folding.png

$\square$

Not every function, however, is an identification map.

Exercise. Given an identification map $f:X\to Y$ and a subset $A\subset X$, show that the restriction $f\Big|_A:A\to f(A)$ doesn't have to be an identification map.

Theorem. Let $q:X\to X/_{\sim}$ be the identification map. Suppose $ $f:X\to Y$ is a map that is constant on each equivalence class of $X$. Then there is a map $f':X/_{\sim}\to Y$ such that $f'q=f$. In other words, there is a map for the bottom arrow to make this diagram commutative: $$ \newcommand{\ra}[1]{\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{cccccccccc} X & & \\ \da{q} & \searrow ^f & \\ X/_{\sim} & \ra{f'} & Y \end{array} $$

What happens to maps when one or both of the spaces are subjected to the quotient construction?

Suppose first that just the domain of a map $f:X\to Y$, is equipped with an equivalence relation. Then quotient map $[f]:X/_{\sim} \to Y$ of $f$ is given by $[f]([x]):=f(x)$.

Quotient of map.png

Of course, the new map is well-defined only if $f$ takes each equivalence class to a single point; i.e., $$x\sim x' \Longrightarrow f(x)=f(x').$$

Exercise. What if, this time, the target space $Y$ has an equivalence relation too? Analyze the possibility of a map $[f]:X \to Y/_{\sim}$.

The general case of a map from a quotient space to a quotient space is familiar from algebra. Given a map $f:X\to Y$, its quotient map $[f]:X/_{\sim} \to Y/_{\sim}$ is given by $$[f]([x]):=[f(x)].$$

Exercise. When is $[f]$ well-defined?

Examples

Example (cylinder). Let's consider the cylinder construction and examine what happens to the topology. The results are similar to the example of the circle: the preimage of an open disk under the identification map is either an open disk or the union of two half-disks at the edge.

Square glue to cylinder with nbhds.png

$\square$

Example (torus). One can construct the torus ${\bf T}^2$ from the cylinder by gluing the top to the bottom: $$(x,0) \sim (x,1).$$ This is how it is visualized:

Torus from cylinder.png

Meanwhile, the equivalence relation for the torus built directly from the square is as follows: $$(0,y) \sim (1,y)\text{ and } (x,0) \sim (x,1).$$ As the torus is a quotient of the square, one can easily see the three types of neighborhoods created by the gluing:

Torus construction.png

In the 3d world, there is a different way to glue the edge to itself:

Torus from cylinder 2.png

$\square$

Example (Klein bottle). One can get the Klein bottle ${\bf K}^2$ from the cylinder by gluing the top to the bottom in reverse: $$(x,0) \sim (1-x,1).$$ The horizontal arrows point in the opposite directions:

Klein bottle construction.png

To bring them together, we need to “cut” through the cylinder's side. This is how it is visualized:

Klein bottle from cylinder.png

Just as with the torus, there are two ways to construct the Klein bottle from the square:

Klein bottle from square.png

$\square$

Exercise. Identify this space:

Sphere construction.png

Example (projective plane). We make the projective plane ${\bf P}^2$ from this square too:

Projective plane.png

One can understand it as if the diametrically opposite, or “antipodal”, points on the boundary of the disk are identified. $\square$

Exercise. Show that we can, alternatively, start with ${\bf R}^2 \setminus \{0,0 \}$ and choose the lines through the origin to be the equivalence classes.

Example (complex projective plane). It is similar, but the reals replaced with the complex numbers: $${\bf CP}^n:={\bf S}^{2n+1} / _{\sim},$$ where $$z\sim z' \Longleftrightarrow z=e^{i\theta}z',\ \exists\theta.$$ and $$\hspace{.3in }{\bf S}^{2n+1}=\{z\in {\bf C}^{n+1}:\ ||z||^2=1\}.\hspace{.3in }\square$$

Exercise. What happens if we identify the antipodal points on the circle ${\bf S}^1$?

Example. There are many ways to create a circle. An insightful way is to make it from the line. One just winds the helix, which is ${\bf R}$ topologically, around the circle, ${\bf S}^1$:

Circle as quotient of R.png

Then the identification map $q : {\bf R} \to {\bf S}^1$ may be thought of as the restriction of the projection of the $3$-space to the plane, or it is given explicitly by $$\hspace{.32in } q(t) := (\cos(\pi t), \sin(\pi t)),\ t \in {\bf R}. \hspace{.32in }\square$$

Example. Suppose we have a room with two doors and suppose as you exit through one door you enter through the other (it's the same door!). If you look through this doorway, this is what you see:

View through quotient door.png

The reason is that, in this universe, light travels in circles... If you run fast enough, an outside observer might see (parts of) you at two different places at the same time:

Quotient of cube outside.png

To produce this effect, we identify the front wall with the back wall. $\square$

Exercise. Describe this situation as a quotient of the cube.

Exercise. What if, as you exit one door, you enter the other -- but upside down? Which of the two below is the correct view?

View through quotient door upside down.png

Exercise. What if the front wall is turned $90$ degrees before it is attached to the back wall? Sketch what you'd see.

Mirrors.png

Exercise. If you've seen two mirrors hung on the opposite walls of the room, you know what they show: you see your face as well as the back of your head, with this pattern repeated indefinitely. Use quotients to achieve this effect without mirrors.

Exercise. Repeat the last exercise for the second, web-cam, image above.

Exercise. What if the room had $n$ walls? What if we have mirrors on all four walls of the room? Five walls, $n$ walls? Four walls, ceiling, and floor?