3-manifold - Foundational Results

Foundational Results

Some results are named as conjectures as a result of historical artifacts.

We begin with the purely topological:

• Moise's theorem – Every 3-manifold has a triangulation, unique up to common subdivision
  • As corollary, every compact 3-manifold has a Heegaard splitting.
• Prime decomposition theorem
• Kneser–Haken finiteness
• Loop and sphere theorems
• Annulus and torus theorem
• JSJ decomposition, also known as the toral decomposition
• Scott core theorem
• Lickorish-Wallace theorem
• Waldhausen's theorems on topological rigidity
• Waldhausen conjecture on Heegaard splittings

Theorems where geometry plays an important role in the proof:

• Smith conjecture
• Cyclic surgery theorem

Results explicitly linking geometry and topology:

• Thurston's hyperbolic Dehn surgery theorem
• The Jørgensen–Thurston theorem that the set of finite volumes of hyperbolic 3-manifolds has order type .
• Thurston's hyperbolization theorem for Haken manifolds
• Tameness conjecture, also called the Marden conjecture or tame ends conjecture
• Ending lamination conjecture