[Paper Review] Higher-order linking forms for 3-manifolds

Constance Leidy|arXiv (Cornell University)|Apr 23, 2012

Geometric and Algebraic Topology9 references4 citations

TL;DR

This paper introduces higher-order linking forms on higher-order Alexander modules for closed, oriented, connected 3-manifolds using coefficient systems in poly-torsion-free-abelian (PTFA) groups and Ore domains. It generalizes classical and knot-based linking forms, and establishes that infection by a knot decomposes the linking form on the resulting manifold as a direct sum of the original and the knot's classical Blanchfield form when the infection curve is non-trivial in homology.

ABSTRACT

Given a closed, oriented, connected 3-manifold, M, we define higher-order linking forms on the higher-order Alexander modules of M. These higher-order linking forms generalize similar linking forms for knots previously studied by the author, which were themselves generalizations of the classical Blanchfield linking form for a knot. We also investigate the effect of the construction known as "infection by a knot" on these linking forms.

Motivation & Objective

• To define higher-order linking forms on higher-order Alexander modules for 3-manifolds using general Ore domains and PTFA coefficient systems.
• To generalize classical and knot-based linking forms, including the Blanchfield form, to the setting of 3-manifolds.
• To investigate the effect of infection by a knot on these higher-order linking forms.
• To show that when the infection curve is non-trivial, the linking form on the infected manifold splits as a direct sum of the original and the classical Blanchfield form of the infecting knot.

Proposed method

• Define higher-order Alexander modules as the torsion submodule of the first homology with coefficients in an Ore domain R, where ZΓ ⊆ R ⊆ KΓ and Γ is a PTFA group.
• Construct the higher-order linking form Bℓ_R as a map from TH_1(M; R) to its dual (TH_1(M; R))#, using the Bockstein sequence and Poincaré duality.
• Use the right ring of fractions KΓ of the group ring ZΓ for coefficient systems, ensuring flatness and allowing duality arguments.
• Apply the infection construction to 3-manifolds by embedding a knot J along a curve η in M, producing a new manifold M(η,J).
• Analyze the induced maps on homology and linking forms via the split short exact sequence in homology arising from infection.
• Prove the linking form on M(η,J) is isomorphic to the direct sum of the linking forms on M and J when φ(η) ≠ 1, using duality and commutative diagrams of maps.

Experimental results

Research questions

• RQ1 How can higher-order linking forms be defined on 3-manifolds using non-localized, general Ore domain coefficients?
• RQ2 What is the behavior of higher-order linking forms under the infection operation on 3-manifolds?
• RQ3 How does the linking form on an infected 3-manifold relate to the linking forms on the original manifold and the infecting knot?
• RQ4 Under what conditions does the linking form on the infected manifold decompose as a direct sum of the original and the classical Blanchfield form?
• RQ5 What role does the non-triviality of the infection curve in homology play in the structure of the resulting linking form?

Key findings

• The higher-order linking form Bℓ_R is well-defined on the torsion submodule of H_1(M; R) for any Ore domain R with ZΓ ⊆ R ⊆ KΓ and PTFA group Γ.
• The linking form generalizes the classical Blanchfield form and previous knot-based higher-order forms to the setting of general 3-manifolds.
• When the infection curve η satisfies φ(η) ≠ 1, the homology of the infected manifold M(η,J) admits a split short exact sequence involving H_1(E(J); R) and TH_1(M; R).
• The linking form on M(η,J) is isomorphic to the direct sum of the linking forms on M and on the infecting knot J, i.e., Bℓ_R(M(η,J)) ≅ Bℓ_R(M) ⊕ Bℓ_R(J).
• The linking form on M(η,J) is completely determined by the linking forms on M and on J, with the classical Blanchfield form of J appearing via a canonical map ψ_*.
• The decomposition result holds precisely when the infection curve is non-trivial in the coefficient system, as shown by the splitting of the homology sequence and the vanishing of cross-terms in the linking form.