Math 2.0

Manifolds and cell complexes Three-manifolds Theorem

Sphere theorem for 3-manifolds

Builds on 3-manifoldOrientabilityHomotopy groupDehn's lemma

Statement

Let MM be an orientable 3-manifold whose second homotopy group π2(M)\pi_2(M) is not trivial. Then there is an embedding S2→MS^2 \to M of the 22-sphere that is not homotopic to a constant map.

Remarks

Papakyriakopoulos proved the theorem in 1957 with the tower construction also used for Dehn's lemma. For example, in M=S2×S1M = S^2 \times S^1 the sphere S2×{1}S^2 \times \set{1} is embedded and generates π2(M)≅ℤ\pi_2(M) \cong \Z. A main consequence: if MM is orientable and irreducible, meaning that every embedded 22-sphere bounds a 33-ball, then π2(M)=0\pi_2(M) = 0. If moreover π1(M)\pi_1(M) is infinite, then MM is aspherical, that is, its universal cover is contractible. Orientability cannot be dropped: in the non-orientable case one must also allow embedded two-sided projective planes, as in Epstein's version of the theorem.

Going deeper

For a student

The theorem explains why irreducible orientable 33-manifolds have trivial π2\pi_2. With infinite fundamental group, they are then Eilenberg–MacLane spaces, determined up to homotopy by π1\pi_1. This is the starting point for the study of 33-manifolds through their fundamental groups.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

Show the path as a graph
3-manifold, concept, no evidence level3-manifoldconceptHomotopy group, concept, no evidence levelHomotopy groupconceptKnot, concept, no evidence levelKnotconceptKnot group, concept, no evidence levelKnot groupconceptOrientability, concept, no evidence levelOrientabilityconceptDehn's lemma, statement, no evidence levelDehn's lemmastatementSphere theorem for 3-manifolds, statement, no evidence levelSphere theorem for 3-…statement
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. 3-manifold concept
  2. Homotopy group concept
  3. Knot concept
  4. Knot group concept
  5. Orientability concept
  6. Dehn's lemma statement
  7. Sphere theorem for 3-manifolds statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Sphere_theorem_(3-manifolds)
About this page
Tag5QAFGPAZ4A
URIm2://5QAFGPAZ4A
Kindstatement, theorem
WritingW0 stub
MSC57K30
StatusThis copy of the site has no build data. Run m2 status 5QAFGPAZ4A.
About3-manifold
Uses3-manifold, Orientability, Homotopy group, Dehn's lemma
Same as
Changed2 November 2026

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