Math 2.0

Manifolds and cell complexes Topological manifolds: basic notions Theorem

Annulus theorem

Builds on Schoenflies theoremTopological embeddingHomeomorphismn-sphereEuclidean space

Statement

Let n≥2n \ge 2. Let Σ1,Σ2⊂ℝn\Sigma_1, \Sigma_2 \subset \R^n be disjoint locally flat embedded (n−1)(n-1)-spheres, that is, images of topological embeddings Sn−1×[−1,1]→ℝnS^{n-1} \times [-1, 1] \to \R^n restricted to Sn−1×{0}S^{n-1} \times \set{0}. For i=1,2i = 1, 2, let UiU_i be the bounded component of ℝn∖Σi\R^n \setminus \Sigma_i, and suppose Σ1⊂U2\Sigma_1 \subset U_2. Then the region U2‾∖U1\overline{U_2} \setminus U_1 is homeomorphic to the annulus Sn−1×[0,1]S^{n-1} \times [0, 1].

Remarks

Each Ui‾\overline{U_i} is a closed ball by the generalized Schoenflies theorem. This does not give the statement, because the inner ball may sit inside the outer one in a complicated way. For n≤3n \le 3 the theorem is classical; in dimension 33 it rests on Moise's work on triangulations. For n≥5n \ge 5 it was proved by Robion Kirby in 1969, using the torus trick, and for n=4n = 4 by Frank Quinn in 1982. An equivalent form is the stable homeomorphism theorem: every orientation-preserving homeomorphism of ℝn\R^n is a finite composition of homeomorphisms that are the identity on some open set. The local flatness hypothesis is necessary: wild spheres such as the Alexander horned sphere give counterexamples in dimension 33.

Going deeper

For a student

The annulus theorem was long called the annulus conjecture, and its proof was a turning point in the theory of topological manifolds. It is needed to show that the connected sum of oriented topological manifolds is well defined. It is also a step in Kirby and Siebenmann's study of triangulations and piecewise linear structures on topological manifolds. Kirby's torus trick, introduced for this proof, became a standard tool of the subject.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Circle, concept, no evidence levelCircleconceptClosed ball, concept, no evidence levelClosed ballconceptConnected space, concept, no evidence levelConnected spaceconceptEuclidean space, object, no evidence levelEuclidean spaceobjectHomeomorphism, concept, no evidence levelHomeomorphismconceptTopological embedding, concept, no evidence levelTopological embeddingconceptn-sphere, object, no evidence leveln-sphereobjectJordan curve theorem, statement, no evidence levelJordan curve theoremstatementSchoenflies theorem, statement, no evidence levelSchoenflies theoremstatementAnnulus theorem, statement, no evidence levelAnnulus theoremstatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Circle concept
  2. Closed ball concept
  3. Connected space concept
  4. Euclidean space object
  5. Homeomorphism concept
  6. Topological embedding concept
  7. n-sphere object
  8. Jordan curve theorem statement
  9. Schoenflies theorem statement
  10. Annulus theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Annulus_theorem
About this page
TagHAT8HG0RCK
URIm2://HAT8HG0RCK
Kindstatement, theorem
WritingW0 stub
MSC57N45, 57N99
StatusThis copy of the site has no build data. Run m2 status HAT8HG0RCK.
AboutTopological embedding
UsesSchoenflies theorem, Topological embedding, Homeomorphism, n-sphere, Euclidean space
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Changed2 November 2026

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