On virtual chirality of 3-manifolds

TL;DR

The paper resolves virtual chirality for compact orientable $3$-manifolds by proving that any prime $3$-manifold not finitely covered by $S^3$ or by a product has a finite chiral cover, and that reducible manifolds inherit chirality from chiral summands. It analyzes hyperbolic, mixed, and graph manifolds via a trio of case-specific strategies that leverage Agol–Wise virtual specialization, slope/Seifert data, and controlled finite coverings to obstruct orientation-reversing self-homeomorphisms. The hyperbolic case uses subgroup-conjugacy-distinguished properties of fundamental groups and trace arguments to preclude reversing isometries; mixed manifolds are handled by assembling covers that force positive boundary-cokernel data and embedding a chiral piece; graph manifolds are treated with slope- and genus-based constructions to defeat global reversals. The reducible case reduces to finding a chiral finite cover of a chiral prime summand and, if necessary, replacing a negative summand by a non-homeomorphic cover to maintain global chirality. Together, these results suggest that virtually all $3$-manifolds are chiral in some finite cover, with explicit constructive procedures for producing such covers.

Abstract

We prove that if a prime 3-manifold M is not finitely covered by the 3-sphere or a product manifold, then M is virtually chiral, i.e. it has a finite cover that does not admit an orientation reversing self-homeomorphism. In general if a 3-manifold contains a virtually chiral prime summand, then it is virtually chiral.

Theorems & Definitions (20)

• Theorem 1.1
• Theorem 1.2
• Proposition 3.1
• Definition 3.2
• Theorem 3.3
• proof
• Remark 3.4
• Proposition 4.1
• Proposition 4.2
• Lemma 4.3