[Paper Review] A Functorial Symplectic Instanton Homology via Traceless Character Varieties

Henry T. Horton|arXiv (Cornell University)|Nov 29, 2016

Geometric and Algebraic Topology3 citations

TL;DR

This paper constructs a functorial symplectic instanton homology, SI(Y), for closed, oriented 3-manifolds using Lagrangian Floer homology in SU(2) traceless character varieties associated to Heegaard splittings. It establishes naturality with respect to Heegaard decompositions, yielding a concrete invariant rather than an isomorphism class, and extends the construction to SU(r) for r > 2, providing a functorial framework for cobordisms between 3-manifolds.

ABSTRACT

Using ideas pioneered by Wehrheim and Woodward, we associate to any closed, oriented $3$-manifold $Y$ a finitely generated abelian group $\mathrm{SI}(Y)$ obtained from (quilted) Lagrangian Floer homology in a certain moduli space of $\mathrm{SU}(2)$-representations associated to a special kind of handlebody decomposition of $Y$. We show that $\mathrm{SI}(Y)$ is natural with respect to Heegaard splittings of $Y$, so that it may be considered as a concrete group as opposed to an isomorphism class of a group. By adapting constructions of Ozsv\'ath and Szab\'o from Heegaard Floer homology to our setting, we show how to obtain functorial invariants of cobordisms between connected $3$-manifolds. We also generalize the construction to the case of $\mathrm{SU}(r)$-representations, $r > 2$.

Motivation & Objective

• To define a concrete, natural symplectic instanton homology invariant SI(Y) for closed, oriented 3-manifolds, avoiding dependence on isomorphism classes.
• To adapt Heegaard Floer homology techniques from Ozsváth and Szabó to the symplectic instanton setting.
• To extend the construction from SU(2) to SU(r) for r > 2, broadening the scope of the invariant.
• To establish functoriality for cobordisms between connected 3-manifolds using this homology theory.
• To provide a geometric and topological framework for instanton homology via moduli spaces of traceless SU(r)-representations.

Proposed method

• Utilize Wehrheim and Woodward’s quilted Lagrangian Floer homology framework to define SI(Y) in the moduli space of SU(2)-representations.
• Construct SI(Y) from a special handlebody decomposition of Y, ensuring naturality under Heegaard splittings.
• Define the homology group as the Lagrangian Floer homology of a pair of Lagrangians associated to the handlebody decomposition.
• Generalize the construction to SU(r)-representations by replacing SU(2) with SU(r) in the character variety and Lagrangian setup.
• Adapt Ozsváth–Szabó’s cobordism maps to the symplectic instanton setting, ensuring functoriality under cobordisms.
• Ensure the resulting invariant is finitely generated abelian and independent of auxiliary choices up to canonical isomorphism.

Experimental results

Research questions

• RQ1 Can a symplectic instanton homology be defined as a concrete group rather than an isomorphism class, using Heegaard splittings?
• RQ2 How can the functoriality of instanton homology under cobordisms be established in the symplectic setting?
• RQ3 To what extent can the SU(2) construction be generalized to SU(r) for r > 2?
• RQ4 What role do traceless character varieties play in defining a well-behaved instanton homology invariant?
• RQ5 How do the techniques of Wehrheim and Woodward’s quilted Floer homology apply to the construction of functorial 3-manifold invariants?

Key findings

• SI(Y) is a finitely generated abelian group that is naturally associated to any closed, oriented 3-manifold Y, independent of choices up to canonical isomorphism.
• The construction is natural with respect to Heegaard splittings, meaning SI(Y) is well-defined as a specific group rather than a class of isomorphic groups.
• Functorial invariants for cobordisms between connected 3-manifolds are obtained by adapting Ozsváth–Szabó’s cobordism map framework to the symplectic instanton setting.
• The construction generalizes to SU(r)-representations for r > 2, extending the invariant to higher-rank structure groups.
• The resulting homology theory is functorial under cobordisms, providing a topological quantum field theory-like structure in the symplectic instanton context.
• The use of traceless character varieties ensures the moduli space is well-suited for Lagrangian Floer homology and captures essential SU(r)-representation data.