Manifold #18. Homotopy매니폴드 #18. 호모토피

These are my notes on XylyXylyX’s What is a Manifold? lectures. This post covers lesson 18 of the playlist, Homotopy, and it is the last post of this series. Up to the last post we examined a space inside a topological space through points and open sets. This post is the first lecture of algebraic topology, and we start reading the curves in a space as a group.

What we need from group theory

The lecture opens by insisting that you know group theory. This post uses roughly the following.

• The definition of a group
• A homomorphism between groups. This is a different thing from a homeomorphism between topological spaces. It need not be one-to-one but it keeps the structure of the group.
• The kernel and normal subgroups of a homomorphism, and isomorphisms

We do not use them right away, but anyone who knows homomorphisms knows these too. A group is an object with more structure than a topological space, so translating properties of a space into the language of groups gives us something to compute.

Curves and homotopy

Pick two points $a$ and $b$ in a topological space $X$. A curve is a continuous function $\gamma : \I \to X$ from a closed interval to $X$, and here we look at curves with $\gamma(0) = a$ and $\gamma(1) = b$. From now on all functions are continuous and I will not say so each time. There can be several curves from the same $a$ to $b$, and we name them $\gamma_0$, $\gamma_1$, and so on.

A homeomorphism continuously deformed a space, open sets and all, into another space. Now we ask whether we can continuously deform one curve into another inside a single space. Each point of the curve moves a little at a time, the curve is never cut, and stretching or shrinking is allowed. If this can be done, the two curves are called homotopic.

In symbols it is a function of two variables.

The variable $s$ is a parameter giving the position on the curve, and $t$ says which curve we are on. Writing $\gamma_t(s) = F(s, t)$, as $t$ runs from 0 to 1 the curve changes gradually from $\gamma_0$ to $\gamma_1$. Every intermediate curve must have the same endpoints, so there is one more condition.

This function $F$ is called a homotopy from $\gamma_0$ to $\gamma_1$. This post treats only homotopies with fixed endpoints. The lecture postpones broader homotopies in which the endpoints may differ.

Finding one homotopy $F$ does not mean we have found every curve homotopic to it. We have only shown that $\gamma_0$ is homotopic to $\gamma_1$, and it may be homotopic to other curves not in the picture too.

In the plane everything is the same, and a hole splits the curves

Any two curves joining two points in the plane $\R^2$ are homotopic to each other. It does not matter if the curves cross. The proof is easy, so the lecture does not show it. One way is to move each point along the straight line joining the two curves. $F(s, t) = (1 - t)\,\gamma_0(s) + t\,\gamma_1(s)$, and the endpoints stay fixed because the two curves share them. (This formula is added for understanding and was not covered in the lecture.) Since all curves are homotopic to each other, the plane is the simplest space in this respect. Homotopy has nothing to find out there.

A convex subset is the same. A set is convex if the segment joining any two of its points always lies in the set, and the straight-line interpolation above does not leave the set, so all curves in it are homotopic. A non-convex set can behave the same way if it has no hole. What causes trouble is a hole.

Consider the space $\R^2 \setminus D$, the plane with a disc $D$ removed. Put the points $A$ and $B$ on either side of the hole, and there is a curve passing above and a curve passing below. There is a hole between them, so we cannot move one to the other without crossing the hole. In the big plane $\R^2$ they are homotopic, but in the space with a hole they are not. Beyond the curves passing above once and below once, a curve that goes once around the hole and on to $B$, and a curve that goes twice around, are not homotopic to any of the others. Each number of turns gives one group of homotopic curves, so there are infinitely many groups.

Check it with the toy below. Drag the vertices of the blue polyline to change the curve, and when the hole is present the toy tells you with a number $k$ whether it is in the same group as the reference curve. A curve touching the hole is not a curve of this space, so it turns red. The slider $t$ shows the curve moving by straight-line interpolation from the reference curve to the blue curve, and if you switch to “No hole” every curve is in one group.

(The number $k$ is fixed by the angle the curve sweeps around the hole. This number and the verdicts of the toy are added for understanding and were not covered in the lecture.)

Q. For a homotopy to exist, must the two curves have the same endpoints?
A. In this post, yes. Every intermediate curve must also start at $a$ and end at $b$. There are homotopies in which the endpoints may move too, but the lecture treats them later.

Seeing this large difference the hole makes, the lecture takes it as a hint that homotopy classes can measure something about a space. Homeomorphic spaces have the same properties regarding homotopy. But two spaces with the same properties of curve classes are not necessarily homeomorphic. The big hope of untangling all of topology with group theory did not come true, and groups tell us only part of topology. Topology and group theory are not subjects in one-to-one correspondence.

Homotopy is an equivalence relation

We check that the relation “is homotopic to” among curves is an equivalence relation. Write $\gamma_0 \simeq \gamma_1$ for homotopic curves.

1. Reflexive: every curve is homotopic to itself. Take $F(s, t) = \gamma(s)$.
2. Symmetric: if $F(s, t)$ is a homotopy from $\gamma_1$ to $\gamma_2$, then $F(s, 1 - t)$ is a homotopy from $\gamma_2$ to $\gamma_1$. It runs the change between the curves backwards.
3. Transitive: if $F$ is a homotopy from $\gamma_0$ to $\gamma_1$ and $H$ is a homotopy from $\gamma_1$ to $\gamma_2$, then $G(s, t)$, which glues the two homotopies, goes from $\gamma_0$ to $\gamma_2$.

We run $t$ twice as fast, putting $F$ in the first half and $H$ in the second. At $t = \tfrac12$ both formulas give $\gamma_1$, so they join.

Since it is an equivalence relation, we can handle the whole collection of homotopic curves without looking at them one at a time. This is the equivalence class $[\gamma]$ of $\gamma$, and any curve of the collection can be put in the brackets as a representative. It is the same idea as grouping equivalence classes in a quotient space. In the plane with a hole there are infinitely many classes, such as the class of curves passing above the hole, the class passing below, and the class going around once more, and in the plane without a hole there is only one class.

Q. Why does it matter that being homotopic is an equivalence relation?
A. Because we can treat a class as a single element without examining curves one by one. If we define a product between classes we can build a group, and this group is the goal of this post.

The product of curves

To reach algebra we must be able to multiply curves. If $\alpha$ goes from $a$ to $b$ and $\beta$ goes from $b$ to $c$, we make the curve $\alpha\beta$ that goes along $\alpha$ first and continues along $\beta$.

Run along $\alpha$ at double speed, then along $\beta$ at double speed, and while $s$ goes from 0 to 1 we travel from $a$ to $c$. This product gives the same class whichever curve of the class of $\alpha$ and whichever curve of the class of $\beta$ we use. So we define the product of classes as follows.

For three curves, $\alpha\beta\gamma$ is glued with $3s$, $3s - 1$, $3s - 2$, and $([\alpha][\beta])[\gamma] = [\alpha]([\beta][\gamma])$ gives associativity. A group needs associativity, so this is good news. There is an inverse path too. If $\beta$ goes from $b$ to $c$, then $\bar\beta(s) = \beta(1 - s)$ goes back from $c$ to $b$. It is the curve that walks the same road backwards.

We almost have the material for a group, but not quite. The product is defined only when the endpoint of the first curve equals the starting point of the second, and every pair of points has its own classes of curves, which is too many. So we narrow the scope.

Loops and the fundamental group

Fix one point $b$ and look only at curves that start at $b$ and return to $b$. These are called loops, and $b$ is called the base point. A loop is a special case of a curve and everything else is the same. A homotopy between loops also fixes the endpoint $b$, and if there is a hole, loops that go around the hole in different ways are not homotopic.

Since the base point is the same, any two loops can be multiplied. The set of all classes of loops has everything a group needs.

• Product: $[\alpha][\beta] = [\alpha\beta]$, which is associative.
• Identity: the class of the constant curve that never leaves $b$.
• Inverse: the inverse of $[\alpha]$ is $[\bar\alpha]$, which goes around in the opposite direction.

This group is called the fundamental group of $X$ at $b$ and is written $\pi_1(X, b)$. It is also called the first homotopy group. Which space and which point the loops start from is all the information, so we write both $X$ and $b$. The lecture says it is the simplest and the most important group. In a plane with one hole, we can count the number of times we go around, so the group has countably infinitely many elements. (The remark that this group equals the additive group of integers is added for understanding and was not covered in the lecture.)

There is one group per base point, which looks burdensome. But within one piece of the space, the fundamental groups at all points are isomorphic to each other. Pick a curve $\delta$ from a point $e$ to a point $h$, and send a loop $\gamma$ at $h$ to $\delta\gamma\bar\delta$. This is a loop at $e$ that goes along $\delta$ to $h$, runs the loop, and returns to $e$ along $\bar\delta$. This correspondence is an isomorphism of groups. (The lecture says “connected region”, but it means two points can be joined by a curve, so precisely the condition is path-connected.) So we may leave out the base point and speak only of the fundamental group $\pi_1(X)$ of the space.

Loops seen as circles

A loop satisfies $\alpha(0) = \alpha(1)$, so it can be seen as a function from the quotient of the interval $\I$ with its two endpoints glued to a single point into $X$. The interval with its endpoints glued is homeomorphic to the circle $S^1$. So a loop is also a function from the circle to $X$. If we write the circle as $e^{2\pi i s}$ in the complex plane, it is natural that $s = 0$ and $s = 1$ are the same point. If there is a hole, this circle may wind around the hole or may miss it.

In the literature one often uses a space $(X, b)$ with one marked point, called a pointed space. A function between pointed spaces usually must send the marked point to the marked point, as in $f(c) = b$.

The homomorphism a continuous function induces

Let $f$ be a continuous function from a pointed space $(Y, c)$ to $(X, b)$ with $f(c) = b$. If $\alpha$ is a loop in $Y$ with base point $c$, then $f \circ \alpha$, which $f$ carries into $X$, is a loop at $b$. So we get a function between fundamental groups.

If two loops are joined by a homotopy and we carry them over with $f$, the carried loops are homotopic too, so this definition does not depend on which loop we choose as the representative. (This check is added for understanding and was not covered in the lecture.) This $f_*$ is called the homomorphism induced by $f$.

It really is a homomorphism. It keeps products, sends inverses to inverses, and sends the constant curve to the constant curve.

This is a big conclusion: a continuous function defined through topology produces a homomorphism of groups. It is a homomorphism and not an isomorphism, so the two fundamental groups need not be the same. A space with two holes has loops around one hole, loops around the other, and loops around both in turn, so its group can be larger than that of a space with one hole. A continuous function between such spaces cannot be a one-to-one correspondence.

But if $f$ is a homeomorphism, then $f_*$ is an isomorphism. (This is because $(f^{-1})_*$, induced by the inverse $f^{-1}$, is the inverse of $f_*$. This reason is added for understanding and was not covered in the lecture.) The fundamental groups of two homeomorphic spaces are isomorphic, and the fundamental groups of spaces joined only by a continuous function are joined by a homomorphism. Even so, getting a connection between the two is already a lot.

Q. If the fundamental groups of two spaces are isomorphic, are the spaces homeomorphic?
A. Not necessarily. Homeomorphic spaces do have isomorphic fundamental groups, but the converse does not hold. The fundamental group is a tool that shows properties of a space such as its holes, and it cannot stand in for the whole topology.

Closing the series

This series began by giving a set a collection of open sets. We sorted that topology by properties such as separation, countability, compactness, and connectedness, made manifolds from spaces that look locally like $\R^n$, and then added a differentiable structure and moved on to tangent spaces, cotangent spaces, and the tangent bundle. We made a cylinder and a Möbius strip with quotient spaces, and last time we looked back at the distance that came before topology. In this final post we gathered the curves in a space into classes by homotopy, gave the classes a product, and obtained a group, the fundamental group. The difference between a space with a hole and a space without one showed up in an algebraic object, a group, and each continuous function produced a homomorphism between groups.

At the end of this lesson the lecture announces that the next lecture will continue with ways to describe the fundamental group more concretely, covering spaces, and some examples. These notes do not follow it that far and close here.