Math 2.0

Manifolds and cell complexes Three-manifolds Theorem

Loop theorem

Builds on 3-manifoldManifold with boundaryFundamental groupNormal subgroupDehn's lemma

Statement

Let MM be a 3-manifold with boundary ∂M\partial M, and let F⊆∂MF \subseteq \partial M be a connected surface. Let NN be a normal subgroup of the fundamental group π1(F)\pi_1(F). Assume that the kernel of the map π1(F)→π1(M)\pi_1(F) \to \pi_1(M) induced by inclusion is not contained in NN. Then there is an embedding g ⁣:(D2,∂D2)→(M,F)g \colon (D^2, \partial D^2) \to (M, F) of the closed disk such that the loop g∣∂D2g|_{\partial D^2} represents an element of π1(F)\pi_1(F) that is not in NN.

Remarks

The version with NN is due to Stallings; Papakyriakopoulos proved the case N={1}N = \set{1} in 1957. The proof uses the same tower of double coverings as the proof of Dehn's lemma. Take N={1}N = \set{1}, and suppose some loop that is essential in FF becomes null-homotopic in MM. Then some embedded essential loop in FF bounds an embedded disk in MM. For example, if MM is a solid torus D2×S1D^2 \times S^1 and FF is its boundary torus, the meridian disk D2×{1}D^2 \times \set{1} is such a disk.

Going deeper

For a student

The loop theorem is the reason why "compressible" can be tested with fundamental groups. A two-sided surface in a 33-manifold is incompressible exactly when its fundamental group injects into that of the manifold. Together with Dehn's lemma and the sphere theorem, it is a basic step in the theory of Haken manifolds.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

Show the path as a graph
3-manifold, concept, no evidence level3-manifoldconceptFundamental group, concept, no evidence levelFundamental groupconceptKnot, concept, no evidence levelKnotconceptKnot group, concept, no evidence levelKnot groupconceptManifold with boundary, concept, no evidence levelManifold with boundaryconceptNormal subgroup, concept, no evidence levelNormal subgroupconceptDehn's lemma, statement, no evidence levelDehn's lemmastatementLoop theorem, statement, no evidence levelLoop 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. 3-manifold concept
  2. Fundamental group concept
  3. Knot concept
  4. Knot group concept
  5. Manifold with boundary concept
  6. Normal subgroup concept
  7. Dehn's lemma statement
  8. Loop theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Loop_theorem
About this page
Tag8YYJBVP68F
URIm2://8YYJBVP68F
Kindstatement, theorem
WritingW0 stub
MSC57K30
StatusThis copy of the site has no build data. Run m2 status 8YYJBVP68F.
About3-manifold
Uses3-manifold, Manifold with boundary, Fundamental group, Normal subgroup, Dehn's lemma
Same as
Changed2 November 2026

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