A ball whose outside is not one

In the plane, separating and straightening come together: a simple closed curve cuts the plane in two, and the whole plane can be bent until the curve is a round circle. The two statements are proved by one argument and are usually believed as one fact. In three dimensions the first survives and the second does not, and the object that shows it was built by James Waddell Alexander in 1924 out of nothing but a pair of horns doing the same thing forever.

The surface is a sphere. Not a sphere with a handle, not a surface with a boundary, not something glued out of pieces — a set homeomorphic to the ordinary two-sphere, sitting in ordinary three-space. Its complement has two pieces, because the separation theorem holds in every dimension. The bounded piece is an ordinary open ball. The unbounded piece is not homeomorphic to the outside of a round sphere, and the difference is visible: there is a loop out there that cannot be pulled tight.

Building it #

Start with a round ball and pull two horns out of it, curving them toward each other so that they clasp — each horn passing once through the loop the other would close if its base were joined up.

Stop there and nothing has happened. The surface is still a round sphere in disguise: the two horns can be shortened, uncurled and reabsorbed, and every step of that is a homeomorphism of the whole of space. Two clasped horns are not an obstruction to anything.

The construction is what happens next. Instead of closing each horn off, grow a pair of smaller horns from each tip, clasped with each other in the same way, at right angles to the pair below. Then do it again on each of those four tips, and again, forever, with the horns shrinking geometrically so that the tips converge.

At every finite stage the result is still a sphere in disguise. Stage two can be undone by undoing stage two’s horns and then stage one’s; so can stage seventeen. The surface is only ever a counterexample in the limit, and this is the first thing to be clear about, because it explains why no picture of it can be a proof of anything.

What the limit adds is the set the tips converge to. The tips at stage $n$ number $2^n$; each stage’s horns are a fixed fraction of the last, so the tip positions form a Cantor set — uncountable, totally disconnected, of measure zero. The limiting surface is the union of every stage’s horns together with that Cantor set of tips. It is compact, it is a sphere, and there is nothing left to unclasp because the unclasping would have to be done in a different order at every point of a Cantor set at once.

Why the outside is not what it looks like #

Draw a loop round one of the first-stage horns, near its base, as the hero figure does.

In the complement of the surface, can that loop be pulled tight? Pulling it tight means moving it continuously, never touching the surface, until it shrinks to a point. Slide it along the horn toward the tip and it reaches the clasp, where the horn’s partner is in the way; the loop cannot pass over the tip because two clasped horns are, locally, two linked circles, and the linking number of two linked circles cannot be changed by any motion that keeps them apart.

At any finite stage the answer is that the loop can be freed — go round the two second-stage horns instead, then round the four third-stage ones, and so on out to whatever stage the construction stopped at, where the horns simply end and the loop slips off. The escape route exists at every finite stage and it gets longer at each one.

In the limit there is no last stage, so the escape route never terminates. That is the whole argument, and it is a good example of a limit changing an answer rather than merely refining it: the loop is contractible at every stage, and not contractible in the limit, because contractibility asks for a finite deformation and the deformations available were getting longer without bound.

So the unbounded complementary region is not simply connected. The outside of a round sphere is: any loop out there is pulled tight by sliding it away from the sphere and shrinking it in the open space beyond. Two spaces one of which is simply connected and the other not are not homeomorphic. Therefore no homeomorphism of space carries Alexander’s sphere to a round one, and the Schoenflies statement fails in three dimensions.

What exactly fails, and what does not #

Four statements, and it is worth separating them because they are routinely run together.

The surface is a sphere. True. There is a homeomorphism from $S^2$ onto it — the horns are grown by a sequence of embeddings whose limit is injective and continuous, and injective continuous maps of compact spaces are homeomorphisms onto their images.

The complement has two pieces. True, by Jordan–Brouwer. Nothing about the construction affects that; separation is exactly the statement that survives.

The inside is a ball. True, and it is worth knowing that it is: the bad behaviour is all on one side. The horns are grown outward, so the bounded region is never pinched; it is the increasing union of ordinary balls and is one itself.

The outside is the outside of a ball. False, which is the whole point.

The asymmetry in the last two is the surprising part. It has a consequence worth stating separately: the horned sphere is a counterexample to a statement about embeddings, not about spaces. The sphere and the round sphere are the same space; the outside and the round outside are not. What differs is not either object but the way one sits inside the other, and topology’s habit of asking after spaces rather than after positions is exactly why the distinction took so long to become visible. The same distinction is what makes knot theory a subject at all: every knot is a circle, and what is being studied is never the circle. A sphere in space has two sides and there is no reason for them to behave alike — the surface is not symmetric under swapping them, and Alexander’s construction breaks the symmetry deliberately by growing everything into one side.

What the construction had to get right #

Three conditions are doing work in the recipe, and dropping any of them destroys it, which is a good way to see what the counterexample is made of.

The horns must shrink. If successive stages did not get smaller, the tips would not converge and the limit would not be a compact set, let alone a sphere. Geometric shrinkage is what makes the limit an embedding.

The clasping must be genuine at every stage. A pair of horns that merely came close without passing through each other could be pulled apart, and the escape route for the loop would close in one step. That is why the linking number is computed in every figure here rather than assumed from the drawing: a picture of two nearly-touching circles and a picture of two clasped ones differ in no property the eye is reliable about.

And every horn must clasp with its own partner rather than with a stranger. The construction’s tree structure depends on the clasping being local — pair by pair — so that the obstruction at each level is exactly one clasp deep. The figures check this too, by computing the linking number of horns from different pairs and requiring it to be zero.

Why the plane’s proof cannot be repeated #

The polygonal argument in the plane approximates a curve by polygons, straightens each one, and passes to the limit. Every step has a three-dimensional analogue except the last.

A polyhedral sphere in space can be straightened — that is the piecewise-linear Schoenflies theorem in dimension three, proved by Alexander himself in the same period, and it is why his counterexample had to be a limit rather than a polyhedron. Approximating the horned sphere by polyhedra therefore gives a sequence of surfaces each of which can be straightened.

The straightening maps do not converge. Each one has to undo one more stage of clasping than the last, and undoing a clasp moves points by an amount that does not shrink with the horn: a loop trapped by a clasp has to be carried right out past the tip, however small the tip is. So the maps’ distortion grows without bound and the limit of the sequence is not a homeomorphism.

That is the precise failure, and it identifies what a repaired theorem needs. It needs a hypothesis forcing the surface to sit tamely — the same locally, at every point, as a flat plane in space. Local flatness is that hypothesis, and with it the Schoenflies statement is true in every dimension, by theorems of Brown and of Mazur and Morse from around 1960. The horned sphere is locally flat at every point except the Cantor set of tips, which is a set of measure zero, and that is enough.

A count that says how bad it is #

There is a way to put a number on the failure, and it is the invariant the loop argument was implicitly using.

The unbounded complement’s fundamental group is not trivial, which is the statement above. What it actually is, is much larger than the picture suggests: it is not the integers, as it would be if the obstruction were a single hole. Escaping the first clasp requires going round the two second-stage horns; escaping those requires the four third-stage ones; the generators multiply at every level, and the group produced is infinitely generated. The complement is not merely non-simply-connected, it is non-simply-connected in an unboundedly complicated way, and no finite list of loops accounts for it.

That is the sharpest form of the contrast with the plane. There, any simple closed curve’s outside has the same fundamental group as a punctured plane — the integers, one generator, counted by the winding number — whatever the curve looks like. Here two sides of one sphere have fundamental groups of different sizes, and the theorem that would have forced them to agree is the one that fails.

The company it keeps #

The horned sphere belongs to a small family of objects whose job is to show that a plausible sentence needs a hypothesis, and its relatives are worth naming because the family resemblance is the lesson.

Antoine’s necklace, from 1921, is the same idea one dimension down: a Cantor set in space whose complement is not simply connected, built as a chain of linked chains of linked chains. Alexander’s sphere can be built by thickening it. The wild arc is an embedded interval in space whose complement is not simply connected — an arc, with no loop in it at all, and still an obstruction.

What all of them have in common is a construction that is innocent at every finite stage and pathological in the limit, with the pathology living on a Cantor set. It is the same shape as the curve of positive area two rungs below, where the construction is a nested sequence whose limit has a property no stage of it has. A limit is not a large finite stage, and almost every counterexample in this corner of the subject is built by exploiting the difference.

What the pictures cannot show #

Nothing here is the horned sphere. Every figure draws a finite stage, and every finite stage is a round sphere in disguise. The object with the property under discussion is the limit of infinitely many of them, and it has no drawing.

The horns are drawn as their cores. A horn is a solid tube and what is drawn is the circle at its centre, stroked to roughly the tube’s width. The linking numbers reported are the linking numbers of those circles, which is what the clasping means, and the surface itself is nowhere in the picture.

A loop that is drawn round one horn is contractible. In the stage-three picture the loop can be slipped off, by going out past three stages of horn and away. The figure’s caption says what the loop cannot do only in the limit, and the argument in the text is what establishes it.

And a Cantor set of tips cannot be drawn either. At stage three there are eight tips; the set that matters is uncountable, and the drawing stops four stages before anything of interest happens.

Where the ladder goes next #

This closes the ladder that began with a spiral corridor and a ray. The sequence of questions was: which side of the curve is a point on, what is a curve allowed to be, is the inside really a disc, does any of it survive an extra dimension, and what breaks when it does. The last answer is that separation survives and taming does not, and that the repair is a hypothesis about how the surface sits rather than about what it is.

The natural continuation is the machinery that made the argument work. The obstruction here was a loop that could not be contracted, which is the fundamental group doing the only thing it was built for, and the reason it could detect the clasp was the linking number. Alongside that sits the question of which spaces are distinguished by such invariants and which are not — two surfaces with the same Euler characteristic and the same orientability are the same surface, and no such classification exists for the ways a sphere can sit inside space.

What is worth carrying away #

The lesson is about the difference between a property and its limit, and it is sharper here than almost anywhere.

Every finite stage of Alexander’s construction has the property that the outside is a ball, and the limit does not. Nothing went wrong at any stage; the property is simply not closed under the limit being taken, and a proof by approximation is exactly a proof that some property is closed under it. The place to look when an approximation argument fails is not the approximations but the quantity that had to stay bounded — here, the distortion of the straightening maps, which grows by a fixed amount at every stage because a clasp costs the same to undo however small it has become.