Sphere Eversion: I · II · III · IV · V · VI

Homotopy is continuous deformation. This volume builds it from the path of Vol. I, and computes the two facts about spheres that Smale’s proof will need.

Vol. I ended with a restatement: an eversion is a path in the space $\operatorname{Imm}(S^2,\mathbb{R}^3)$. A path in a space of maps is a continuously varying family of maps. That has a name.

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homotopy

The notation $X\times[0,1]$ matters. For $X=S^2$ it is a thickened sphere, a spherical shell: each point of the sphere grows a little time-interval. It is not the solid ball. (The Phi-3 transcript got this wrong, and it is worth getting right: the homotopy parameter is a new direction, not a filling-in.)

In $\mathbb{R}^n$ every two maps are homotopic, by the straight-line homotopy

\[H(x,s)=(1-s)\,f(x)+s\,g(x).\]

The same formula works in any convex set, because the segment between $f(x)$ and $g(x)$ stays inside. Homotopy only becomes interesting when the target has a hole for the segment to fall into. Below, $\gamma_{0}$ and $\gamma_{1}$ are two paths from $A$ to $B$ and the orange curve is the frame $H_{s}$.

$H(t,s)=(1-s)\,\gamma_{0}(t)+s\,\gamma_{1}(t)$, with both endpoints held fixed. In the plane this always works. With the origin removed it works exactly when the loop "$\gamma_{0}$ then $\gamma_{1}$ backwards" has winding number $0$ around the hole. Drag the green handle across the hole and watch both facts change together.

When the endpoints are held fixed throughout, as here, the homotopy is called a homotopy rel endpoints. That is the right notion for paths; without it, any path could be reeled in to a constant one.

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loops and the fundamental group

A loop at $x_{0}$ is a path that starts and ends at $x_{0}$. Two loops can be concatenated: run the first at double speed, then the second.

Checking the group laws is a matter of reparametrizing time, and every reparametrization is itself a homotopy. A space in which every loop is homotopic to a constant (and which is path-connected) is simply connected: $\pi_{1}=0$. Every convex set is simply connected, by the straight-line homotopy.

The widget above shows the first nontrivial example. In $\mathbb{R}^2\setminus\lbrace 0\rbrace$, the loop “$\gamma_{0}$ then $\gamma_{1}$ backwards” is homotopic to a constant exactly when its winding number is $0$. To prove that, we need to define winding number properly, and the cleanest place to do that is the circle.

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the circle: π₁(S¹) = ℤ

The map

\[p:\mathbb{R}\to S^1,\qquad p(s)=(\cos 2\pi s,\ \sin 2\pi s)\]

wraps the line around the circle infinitely often. It is a covering map: every small arc $U$ of the circle has a preimage that is a disjoint union of copies of $U$ (one per integer), each mapped homeomorphically onto $U$. Picture the line as a helix sitting over the circle, with $p$ as vertical projection.

Path lifting. For every path $\gamma$ in $S^1$ starting at $p(0)$ there is exactly one path $\tilde\gamma$ in $\mathbb{R}$ with $\tilde\gamma(0)=0$ and $p\circ\tilde\gamma=\gamma$.

Why. By compactness of $[0,1]$ (Vol. I), chop time into finitely many pieces, each of which $\gamma$ maps into one small arc. Over a small arc, the covering is a stack of copies, and you have no choice but to stay on the copy you started in. Glue the pieces. $\square$

Homotopy lifting. The same argument, applied to a square $[0,1]\times[0,1]$ instead of an interval, lifts homotopies.

Now define the degree of a loop $\gamma$ at $p(0)$ to be $\tilde\gamma(1)$. It is an integer, because $p(\tilde\gamma(1))=\gamma(1)=p(0)$. A homotopy of loops lifts to a homotopy of lifts, along which the endpoint moves continuously in $p^{-1}(p(0))=\mathbb{Z}$. By Vol. I’s corollary, it cannot move. So:

\[\deg:\pi_1(S^1)\xrightarrow{\ \cong\ }\mathbb{Z}.\]

(It is onto because $t\mapsto p(nt)$ has degree $n$, and one-to-one because two loops with the same lift endpoint have lifts related by a straight-line homotopy in the convex set $\mathbb{R}$, which projects down.)

The helix is $\mathbb{R}$, coiled so that $p(s)=(\cos 2\pi s,\sin 2\pi s)$ is "drop straight down onto the circle". The blue point runs a loop $\theta(t)=2\pi\bigl(nt+w\sin 2\pi t\bigr)$ on the circle; the orange point is its unique lift starting at $0$. However much you wiggle, the lift ends exactly on the integer $n$. That integer is the class of the loop in $\pi_{1}(S^1)$. Drag to turn the picture.

The winding number of a loop $\gamma$ in $\mathbb{R}^2\setminus\lbrace 0\rbrace$ is the degree of $\gamma/\lVert\gamma\rVert$, a loop on the unit circle. The straight-line homotopy $(1-s)\gamma+s\,\gamma/\lVert\gamma\rVert$ never hits $0$, so $\mathbb{R}^2\setminus\lbrace 0\rbrace$ and $S^1$ have the same loops up to homotopy: $\pi_{1}(\mathbb{R}^2\setminus\lbrace 0\rbrace)\cong\mathbb{Z}$. That is the missing proof for the first widget.

Keep this pattern in mind; it is the whole of Vol. V in miniature. A covering map $E\to B$ with discrete fibre $F$ (here $\mathbb{R}\to S^1$, fibre $\mathbb{Z}$), plus the fact that $E$ has no interesting loops, computes the loops of $B$ from the points of $F$.

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the sphere: π₁(S²) = 0

Now go up a dimension. Take a loop on $S^2$. If it misses some point $q$, then it lives in $S^2\setminus\lbrace q\rbrace$, and stereographic projection from $q$ is a homeomorphism $S^2\setminus\lbrace q\rbrace\to\mathbb{R}^2$. In $\mathbb{R}^2$ the straight-line homotopy shrinks the loop to a point. Project back.

A loop drawn around the removed south pole. On $S^2\setminus\lbrace S\rbrace$ it slides up over the equator and shrinks to a point at the top: the sphere minus a point is a plane in disguise, and every loop in it is contractible. Remove the north pole too and the space becomes a cylinder; the same loop is caught between the two holes.

There is a real gap in that argument, and it is worth seeing. A continuous loop can hit every point of $S^2$: space-filling curves exist. The repair is to first homotope the loop to a smooth one (approximate it by a smooth loop close enough that the straight-line homotopy between them, pushed back onto the sphere, is well defined). A smooth map from a 1-dimensional thing into a 2-dimensional thing has an image of area zero (this is the easy case of Sard’s theorem), so it misses a point. Then stereographic projection finishes. Therefore

\[\pi_1(S^2)=0.\]

Compare the circle: there the loop that goes once around cannot miss a point, and it is exactly the loop that does not shrink.

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higher homotopy groups

A loop is a map from $S^1$. Replace $S^1$ by $S^k$ and you get the higher homotopy groups:

The argument for $\pi_{1}(S^2)=0$ works word for word in every dimension where the domain is smaller than the target:

Proof. Homotope $f:S^k\to S^n$ to a smooth map. Since $k<n$, its image has $n$-dimensional measure zero (Sard), so it misses a point $q$. Stereographic projection from $q$ carries $S^n\setminus\lbrace q\rbrace$ to $\mathbb{R}^n$, where the straight-line homotopy contracts $f$ (to the base point, if we contract toward it). $\square$

We will use exactly one instance of this, in Vol. V:

\[\pi_2(S^3)=0.\]

A 2-sphere inside a 3-sphere always has room to shrink. When the dimensions match, the answer is instead $\pi_{n}(S^n)\cong\mathbb{Z}$, the degree again: how many times, with sign, the map covers the target. We met the $n=1$ case above; the $n=2$ case is the degree of the Gauss map in Vol. IV. And a warning, to show the theorem above is not “obvious”: when the domain is bigger, things are wild. $\pi_{3}(S^2)\cong\mathbb{Z}$, generated by the Hopf fibration, a map from the 3-sphere onto the 2-sphere that no homotopy can undo.

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where this goes

Two computations to carry forward, and one principle:

1. $\pi_{1}(S^1)=\mathbb{Z}$, via the covering $\mathbb{R}\to S^1$. It will reappear as the turning number of a plane curve in Vol. IV.
2. $\pi_{2}(S^3)=0$, via Sard and stereographic projection. It will be the last link in the proof in Vol. V.
3. Homotopy invariants are discrete, and discrete things cannot move continuously. That is Vol. I’s corollary, and every obstruction in this series is an instance of it.

But nothing so far knows about smoothness. Homotopy lets a sphere be crushed to a point, folded flat, and unfolded. To say what is forbidden in an eversion we need derivatives, and a sphere to take them on.

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Next: Vol. III: Manifolds and Immersions

Sphere Eversion: Vol. I · Vol. II · Vol. III · Vol. IV · Vol. V · Vol. VI