Sphere Eversion: I · II · III · IV · V · VI
One theorem, from the ground up: a sphere can be turned inside out, smoothly, if it is allowed to pass through itself. We start with the definition of an open set.
Everything is built from definitions, and most figures are equations you can drag. The prerequisites are multivariable calculus (partial derivatives, the cross product, the divergence theorem) and a first course in linear algebra (rank, determinant, orthonormal bases). Topology and abstract algebra are not assumed; the one algebraic word the series needs, group, is defined where it first appears. Along the way, boxed questions mark the places where it is instructive to stop and ask, and the exposition after each one answers it.
The animation above is a genuine eversion: Thurston’s corrugations, as rendered in the 1994 film Outside In. By the end of Vol. VI you will know exactly what it is a picture of, and why its existence was proved before anyone could draw it.
| Vol. | Topic | What it contributes to the proof |
|---|---|---|
| I | Open, closed, clopen | Spaces, continuity, connectedness: the reason an integer that varies continuously cannot jump. |
| II | Paths and homotopy | Homotopy, $\pi_{1}(S^1)=\mathbb{Z}$, and $\pi_{k}(S^n)=0$ for $k<n$. |
| III | Manifolds and immersions | Charts, the Jacobian, immersions, regular homotopy, and a homotopy that cheats. |
| IV | Curves in the plane | Turning number and Whitney–Graustein: why a circle cannot evert. |
| V | Frames and Smale's theorem | Fibrations, $SO(3)\cong\mathbb{RP}^3$, the belt trick, and $\pi_{2}(V_{3,2})=0$. |
| VI | Seeing the eversion | Corrugations, a ruled formula for the halfway model, and how to tell you are halfway. |
What is proved, and what is explained. Every theorem the argument uses is either proved on the page or, where a proof is beyond the scope of a blog, stated with each of its objects defined, so that you know exactly what it claims. The existence proof rests on one result of the second kind: the hard half of Smale’s theorem, that a vanishing obstruction really does produce a regular homotopy. Vol. V explains what it says, object by object, and why it is hard, and Vol. VI shows the path it promises. The other results stated without proof (the hairy ball theorem, Hopf’s degree theorem, Gauss–Bonnet, the general long exact sequence, the Hopf fibration, immersions in $\mathbb{R}^4$) are asides, explained where they appear; nothing in the argument leans on them.
The question
Here is the target, stated once in full so that every word can be earned later. Let $\iota(p)=p$ be the standard sphere and $\alpha(p)=-p$ its antipodal copy, whose inside faces out.
“Immersion” is Vol. III. This volume is about the words space and path-connected.
Open balls and open sets
Distance in $\mathbb{R}^n$ is $d(p,q)=\lVert p-q\rVert$. The open ball of radius $\varepsilon>0$ about $p$ is
\[B(p,\varepsilon)=\{\,q\in\mathbb{R}^n:\ d(p,q)<\varepsilon\,\}.\]Open means every point has room. The open disk $\lbrace\lVert p\rVert<1\rbrace$ is open: a point at distance $d<1$ from the centre can use $\varepsilon=1-d$. The closed disk $\lbrace\lVert p\rVert\le 1\rbrace$ is not: a point on the edge belongs to the set, but every ball around it pokes outside.
Equivalently, $C$ is closed when it contains every point it can get arbitrarily close to (its limit points). Closed is not the opposite of open. A set can be both, or neither. Try all three:
Solid edge: those points belong to the set. Dashed edge: they do not. The gold disk is the biggest ball around your point that stays on one side. On the edge no ball works, and which side fails decides whether the set is open, closed, or neither.
The half-open disk is neither: its top edge has points in the set with no room (so it is not open), and its bottom edge has points outside the set with no room in the complement (so it is not closed). On the real line the same thing happens with $[0,1)$.
Two facts about open sets in $\mathbb{R}^n$ are worth checking by hand, because they become the definition of everything else:
- Any union of open sets is open. If $p$ lies in the union it lies in one of the sets, and that set’s ball works.
- A finite intersection of open sets is open. If $p\in U_{1}\cap\dots\cap U_{k}$, take the smallest of the $k$ radii. With infinitely many sets the smallest radius may be $0$: $\bigcap_{n}(-\tfrac1n,\tfrac1n)=\lbrace 0\rbrace$, which is not open.
Topologies
Those two facts, plus “$\varnothing$ and the whole space are open”, are all we ever use. So we promote them to a definition and forget the distance.
Examples that will matter:
- Metric topologies. Any set with a distance function, open sets defined by balls exactly as above. A distance function is a rule $d(p,q)\ge 0$ that is $0$ only when $p=q$, is symmetric, and obeys the triangle inequality $d(p,r)\le d(p,q)+d(q,r)$. This includes $\mathbb{R}^n$, the sphere $S^2$ (with distance measured in $\mathbb{R}^3$), and spaces of maps. For maps $f,g:S^2\to\mathbb{R}^3$ with continuous derivatives (the space $C^1(S^2,\mathbb{R}^3)$), $d(f,g)=\sup\lVert f-g\rVert+\sup\lVert df-dg\rVert$ is a distance, where $df$ is the derivative of $f$ along the sphere, made precise in Vol. III. Two surfaces are close when their points and their tangent planes are close. That is the topology in which Smale’s theorem lives.
- The discrete topology. Every subset is open. On $\mathbb{Z}$ with its usual distance this is what you get, since $B(n,\tfrac12)=\lbrace n\rbrace$.
- The subspace topology. If $Y\subseteq X$, a set is open in $Y$ when it is $U\cap Y$ for some $U$ open in $X$. Openness is relative to the ambient space.
Clopen sets and connectedness
Theorem. The interval $[0,1]$ is connected.
Proof. Suppose $A\subseteq[0,1]$ is clopen and contains $0$; we show $A=[0,1]$. Let $s=\sup\lbrace x: [0,x]\subseteq A\rbrace$, the least upper bound of that set: the smallest number that is $\ge$ every element of it. It need not belong to the set (the least upper bound of $[0,1)$ is $1$), and every nonempty set of reals with an upper bound has one, because the real line has no gaps. Because $A$ is closed, it contains the limit point $s$, so $[0,s]\subseteq A$. Because $A$ is open (in $[0,1]$), if $s<1$ some interval $[s,s+\varepsilon)$ also lies in $A$, contradicting the choice of $s$. So $s=1$ and $A=[0,1]$. If instead $0\notin A$, apply the argument to the complement, which is also clopen. $\square$
This is the completeness of the real numbers, wearing a topological hat. It is the single fact on which the rest of the series leans.
Continuity
In calculus, $f$ is continuous if for every $\varepsilon$ there is a $\delta$. In topology the same idea is one line.
For $f:\mathbb{R}\to\mathbb{R}$ this is exactly $\varepsilon$–$\delta$: take $V=(f(x)-\varepsilon,\,f(x)+\varepsilon)$; “the preimage is open” says some $(x-\delta,\,x+\delta)$ maps into $V$. Drag the interval below and try to find an open $V$ whose preimage is not open.
Hollow dot: endpoint not included. Filled dot: endpoint included. For the continuous function every preimage is a union of open intervals, wherever you put $V$. For the step function some positions of $V$ produce a preimage with a filled endpoint, a point with no room, and that failure is the discontinuity.
A continuous bijection whose inverse is also continuous is a homeomorphism. Homeomorphic spaces are the same space with the points renamed: open sets on one side match open sets on the other, so no topological question can tell them apart. Stretching $(0,1)$ onto $(0,2)$ is a homeomorphism, and so is $x\mapsto x/(1-\lvert x\rvert)$ from $(-1,1)$ onto all of $\mathbb{R}$, with inverse $y\mapsto y/(1+\lvert y\rvert)$.
Now the payoff, in three lines.
Theorem. The continuous image of a connected space is connected.
Proof. If $f:X\to Y$ is continuous and onto (we may take $Y$ to be the image) and $B\subseteq Y$ is clopen, then $f^{-1}(B)$ is clopen in $X$ (preimages respect complements). If $X$ is connected, $f^{-1}(B)$ is $\varnothing$ or $X$, so $B$ is $\varnothing$ or $Y$. $\square$
Corollary (intermediate value theorem). A continuous $f:[0,1]\to\mathbb{R}$ takes every value between $f(0)$ and $f(1)$. (Otherwise a missed value $c$ splits the image into the clopen pieces below and above $c$.)
Corollary (integers cannot jump). Every continuous map from a connected space to $\mathbb{Z}$ is constant. $\mathbb{Z}$ is discrete, so each $\lbrace n\rbrace$ is clopen, and a connected image must be one point.
That last corollary is the engine of the whole subject. In Vol. II the integer is a winding number. In Vol. IV it is the turning number of a curve, and it proves a circle cannot be turned inside out in the plane. In Vol. V the integer is replaced by an element of a group, and the group turns out to be zero.
Paths and path-connectedness
Path-connected spaces are connected: a clopen split of $X$ would pull back, along a path joining the two pieces, to a clopen split of $[0,1]$. The converse fails for exotic spaces. The topologist’s sine curve, the graph of $\sin(1/x)$ for $0<x\le 1$ together with the segment $\lbrace 0\rbrace\times[-1,1]$ that it oscillates toward, is connected, yet no path reaches the segment from the graph. Nothing in this series is that wild. A space is locally path-connected if every point has arbitrarily small path-connected neighbourhoods; manifolds and the spaces of maps we care about are, and for such spaces connected and path-connected agree.
The path components of $X$ are its maximal path-connected pieces; the set of them is written $\pi_{0}(X)$. Now the theorem can be restated with no vague words left except “immersion”:
\[\text{Smale:}\qquad \pi_0\bigl(\operatorname{Imm}(S^2,\mathbb{R}^3)\bigr)=\{\ast\}.\]A point of this space is an entire immersed sphere. A path in it is a movie of immersed spheres, changing continuously in both position and tangent planes. An eversion is a path from $\iota$ to $\alpha$.
Compactness, and why immersions have room
One more word, needed here and again in Vol. II.
$S^2$ is closed and bounded, so compact. The fact we need is: a continuous real function on a compact space attains its minimum. Apply it to an immersion $f$. At each point, its derivatives $f_{u},f_{v}$ along two perpendicular unit tangent directions are independent, and the length $\lVert f_{u}\times f_{v}\rVert$ (the factor by which $f$ stretches area) does not depend on which pair you pick. So $m=\min\lVert f_{u}\times f_{v}\rVert>0$. The cross product is continuous in the derivatives, so any map $g$ whose derivatives are uniformly close enough to $f$’s (how close depends only on $m$ and on the size of $df$) still has $\lVert g_{u}\times g_{v}\rVert>0$. So every immersion has a ball of immersions around it:
\[\operatorname{Imm}(S^2,\mathbb{R}^3)\ \text{is an open subset of}\ C^1(S^2,\mathbb{R}^3).\]On a non-compact surface the infimum could be $0$, approached at infinity, and there would be no room. On the sphere there always is. This is the fact that makes wiggling an immersion safe, and wiggling is how eversions are built.
Compactness gives one more fact of the same kind, used in Vol. II: a continuous map on a compact space is uniformly continuous. For each $\varepsilon>0$ a single $\delta>0$ works at every point at once. (If not, there would be pairs of points closer and closer together whose images stay $\varepsilon$ apart; compactness lets those pairs pile up at one point, and continuity fails there.)
(The last axiom we need is Hausdorff: distinct points have disjoint open neighbourhoods, so limits are unique. Every metric space is Hausdorff. It shows up in the definition of a manifold in Vol. III and nowhere else.)
Where this goes
We now have the language to ask the question. $\operatorname{Imm}(S^2,\mathbb{R}^3)$ is a topological space. Its path components are what we want to count. The tool for counting path components of spaces of maps is homotopy, and the first thing it counts is how many times a loop winds around a hole.