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Alexander horned sphere

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A pathological embedding of the 2-sphere into 3-dimensional Euclidean space

Alexander horned sphere<br>The Alexander horned sphere is a pathological embedding of the 2-sphere into 3-dimensional Euclidean space. The topological object was discovered by J. W. Alexander (1924).

It is a particular topological embedding of a two-dimensional sphere in three-dimensional space. Together with its inside, it is a topological 3-ball, the Alexander horned ball , and so is simply connected; i.e., every loop can be shrunk to a point while staying inside. However, the exterior is not simply connected, unlike the exterior of the usual round sphere.

The Schoenflies theorem in 2D states that any simple closed curve in the plane can be extended to a homeomorphism of the entire plane. If this held in 3D, the exterior of any embedded sphere would have to be homeomorphic to the exterior of a standard sphere. The Alexander horned sphere proves this is false because the exterior of a standard sphere is simply connected, while

{\displaystyle D_{ext}}

is not. Therefore, there is no homeomorphism of

{\displaystyle \mathbb {R} ^{3}}

that can "straighten" the horned sphere into a standard sphere.

History<br>[edit]

In the late 19th century, the Jordan curve theorem established that every simple closed curve in the plane divides it into two regions. Mathematicians of the era, including Camille Jordan and Arthur Moritz Schoenflies, sought to generalize this to higher dimensions. The Schoenflies theorem successfully proved that in two dimensions, any such curve is "well-behaved"—meaning the regions it bounds are always homeomorphic to standard disks. It was widely conjectured that a similar principle would apply to a 2-sphere embedded in 3-dimensional space.

Antoine's necklace<br>[edit]

The conceptual groundwork for structures was laid by the French mathematician Louis Antoine. In 1921, Antoine constructed a remarkable topological object known as Antoine's necklace, a Cantor set in

{\displaystyle \mathbb {R} ^{3}}

whose complement is not simply connected.

Antoine's construction used a sequence of interlocking solid tori. This proved that a "zero-dimensional" object (a Cantor set) could be embedded in 3D space in a way that "snags" loops, a phenomenon impossible in 2D.

Alexander's counterexample<br>[edit]

In 1924, James Waddell Alexander II published his findings in the Proceedings of the National Academy of Sciences. Prior to this, Alexander had actually published a "proof" that the Schoenflies theorem did hold in 3D. Upon realizing his error, he constructed the horned sphere as a definitive counterexample.

Alexander's genius was in realizing that the "wildness" Antoine had found in a Cantor set could be incorporated into the surface of a 2-sphere. By replacing the simple links of Antoine's necklace with "horns" that grow from a sphere, Alexander showed that:

A sphere can be topologically "simple" (homeomorphic to

{\displaystyle S^{2}}

).

Yet, it can be "wildly" embedded such that the exterior space is knotted infinitely many times.

Alexander's discovery forced topologists to distinguish between tame embeddings and wild embeddings.

This distinction led to the development of PL topology (Piecewise Linear) and eventually helped frame the Generalized Poincaré conjecture.

Construction<br>[edit]

Diagram of the first few iterative steps in the construction of Alexander's horned sphere, from Alexander's original 1924 paper<br>The construction of the Alexander horned sphere is an iterative process.

The Alexander horned sphere is the particular (topological) embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting with a standard torus:[1]

Remove a radial slice of the torus.

Connect a standard punctured torus to each side of the cut, interlinked with the torus on the other side.

Repeat steps 1–2 on the two tori just added ad infinitum.

Animated construction of Alexander's sphere.<br>The Alexander horned sphere is the limit of this process as the number of iterations

{\displaystyle n}

approaches infinity (

{\displaystyle n\to \infty }

).

By considering only the points of the tori that are not removed at some stage, the result is an embedding of the sphere with a Cantor set removed.

This embedding extends to a continuous map from the whole sphere, which is injective (hence a topological embedding since the sphere is compact) since points in the sphere approaching two different points of the Cantor set will end up in different 'horns' at some stage and therefore have different images.

The resulting boundary is a continuous surface. Because the horns become infinitely...