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A new proof of the virtual Haken conjecture

Charalampos Charitos

TL;DR

The paper addresses whether every compact, orientable, irreducible 3-manifold with infinite fundamental group has a finite cover that is Haken. It pursues a direct geometric/topological proof by leveraging Thurston's hyperbolization theory, the Cooper–Long–Reid framework for manifolds with boundary, and Dehn surgery on hyperbolic link complements to construct finite covers supporting incompressible surfaces. A key construction is a special surface in the link-complement setting, together with a finite covering that makes this surface separating and incompressible, which either yields a Haken finite cover or forces a surface-bundle structure that still produces a Haken cover. The result provides a purely geometric/topological verification of the virtual Haken property, bridging classical Thurston theory with contemporary 3-manifold topology and offering a pathway distinct from the analytic proofs tied to Perelman’s geometrization.

Abstract

A new direct proof of the Virtual Haken Conjecture, which asserts that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group has a finite cover that is Haken, will be given.

A new proof of the virtual Haken conjecture

TL;DR

The paper addresses whether every compact, orientable, irreducible 3-manifold with infinite fundamental group has a finite cover that is Haken. It pursues a direct geometric/topological proof by leveraging Thurston's hyperbolization theory, the Cooper–Long–Reid framework for manifolds with boundary, and Dehn surgery on hyperbolic link complements to construct finite covers supporting incompressible surfaces. A key construction is a special surface in the link-complement setting, together with a finite covering that makes this surface separating and incompressible, which either yields a Haken finite cover or forces a surface-bundle structure that still produces a Haken cover. The result provides a purely geometric/topological verification of the virtual Haken property, bridging classical Thurston theory with contemporary 3-manifold topology and offering a pathway distinct from the analytic proofs tied to Perelman’s geometrization.

Abstract

A new direct proof of the Virtual Haken Conjecture, which asserts that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group has a finite cover that is Haken, will be given.

Paper Structure

This paper contains 9 sections, 18 theorems, 19 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Thurston's great theorems
  3. Hyperbolic manifolds
  4. The atoroidality condition
  5. The virtual Haken conjecture for 3-manifolds with boundary
  6. The complement of links in hyperbolic 3-manifolds
  7. A special surface in the complement of the handlebody corresponding to link $\mathcal{L}$
  8. Construction of the appropriate covering space
  9. Proof of the virtual Haken conjecture

Key Result

Theorem 1

Let $M$ be a closed orientable manifold. Then $M$ admits a hyperbolic structure provided that: $(1)$$M$ is irreducible and Haken; $(2)$$M$ is atoroidal i.e. $\pi _{1}(M)$ does not contain a subgroup isomorphic to $\mathbb{Z}\oplus \mathbb{Z}.$

Figures (10)

  • Figure 1: The curves $d_{i}.$
  • Figure 2: The curve $c$ which bounds a disc in $N.$
  • Figure 3: The graph $G^{\prime }$ that extends the spine $G$ of handlebody $H.$
  • Figure 4: Step 1 for constructing the surface $Q.$
  • Figure 5: Step 2 for constructing the surface $Q.$
  • ...and 5 more figures

Theorems & Definitions (26)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Conjecture 5
  • Definition 6
  • Theorem 7
  • Definition 8
  • Theorem 9
  • Theorem 10
  • ...and 16 more