[Paper Review] Finiteness of 3-manifolds associated with non-zero degree mappings

Michel Boileau, J Rubinstein+1 more|arXiv (Cornell University)|Nov 22, 2005

Geometric and Algebraic Topology31 references4 citations

TL;DR

This paper establishes finiteness results for 3-manifolds dominated by a given compact, orientable 3-manifold via non-zero degree maps. Using the Thurston norm and JSJ decomposition, it proves that such manifolds have only finitely many homeomorphism types of JSJ pieces, and that a closed 3-manifold dominates only finitely many integral homology spheres and knot exteriors in S³.

ABSTRACT

We prove a finiteness result for the $\partial$-patterned guts decomposition of all 3-manifolds obtained by splitting a given orientable, irreducible and $\partial$-irreducible 3-manifold along a closed incompressible surface. Then using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds dominated by a given compact 3-manifold belong, up to homeomorphism, to a finite collection of compact 3-manifolds. We show also that any closed orientable 3-manifold dominates only finitely many integral homology spheres and any compact 3-manifolds orientable 3-manifold dominates only finitely many exterior of knots in $S^3$.

Motivation & Objective

• To resolve the finiteness problem of 3-manifolds dominated by a given compact, orientable 3-manifold via non-zero degree maps.
• To extend previous results on 1-domination to general non-zero degree maps, particularly for irreducible 3-manifolds not supporting S³, PSL₂(R)~ or Nil geometry.
• To establish finiteness of JSJ-pieces and homology sphere targets under domination, using the Thurston norm and geometric decomposition.
• To prove that the fundamental group of a compact 3-manifold admits non-degenerate homomorphisms to only finitely many knot groups.
• To provide a criterion for non-degenerate homomorphisms between knot groups based on longitude images.

Proposed method

• Analyzes the ∂-patterned guts decomposition of 3-manifolds obtained by splitting along incompressible surfaces.
• Applies the Thurston norm to control the complexity of surfaces and bound the number of JSJ pieces.
• Uses the JSJ decomposition to split irreducible 3-manifolds into Seifert fibered or atoroidal pieces, leveraging geometrization.
• Reduces the finiteness problem to the case of manifolds with non-trivial JSJ decomposition, leveraging known finiteness for hyperbolic and Seifert geometries.
• Translates topological domination into group-theoretic conditions, focusing on non-degenerate homomorphisms between fundamental groups.
• Establishes a criterion for non-degenerate homomorphisms between knot groups using the image of the preferred longitude.

Experimental results

Research questions

• RQ1 Does a closed orientable 3-manifold dominate only finitely many closed, irreducible, orientable 3-manifolds not supporting S³, PSL₂(R)~ or Nil geometry?
• RQ2 Are the JSJ-pieces of all 3-manifolds dominated by a given compact 3-manifold, up to homeomorphism, finite in number?
• RQ3 Does a compact orientable 3-manifold dominate only finitely many integral homology spheres?
• RQ4 Does a compact orientable 3-manifold dominate only finitely many knot exteriors in S³?
• RQ5 Which homomorphisms between knot groups arise from non-zero degree maps, and how can they be characterized?

Key findings

• The JSJ-pieces of all 3-manifolds dominated by a given compact, orientable, irreducible, ∂-irreducible 3-manifold belong to a finite set of homeomorphism types.
• A closed orientable 3-manifold dominates only finitely many integral homology spheres.
• A compact orientable 3-manifold dominates only finitely many knot exteriors in S³.
• The fundamental group of a compact orientable 3-manifold admits non-degenerate homomorphisms to only finitely many distinct knot groups.
• A homomorphism between knot groups is non-degenerate if and only if it maps the preferred longitude to a non-trivial peripheral element.
• The proof relies on the Thurston norm and the fact that geometrizable 3-manifolds with non-trivial JSJ decomposition are controlled by finitely many geometric types.