[Paper Review] Dehn surgery formula for Seiberg-Witten invariants of homology 3-spheres

Weimin Chen|arXiv (Cornell University)|Mar 16, 1997

Geometric and Algebraic Topology8 references10 citations

TL;DR

This paper establishes a Dehn surgery formula that relates the Seiberg-Witten invariants of a homology 3-sphere to its Casson invariant via the structure of its Seiberg-Witten Floer homology. By leveraging the surgery exact triangle and duality properties in monopole Floer homology, the authors derive a precise algebraic formula expressing the Casson invariant as a signed count of certain generators in the Floer homology, confirming a conjecture by Kronheimer and Mrowka.

ABSTRACT

Recently, Kronheimer and Mrowka [KM2] conjectured a formula relating the Casson’s invariant of an oriented homology 3-sphere and its Seiberg-Witten Floer homology. More precisely, let Y be an oriented homology 3-sphere and bound a smooth compact spin 4-

Motivation & Objective

• To verify a conjecture by Kronheimer and Mrowka relating the Casson invariant of a homology 3-sphere to its Seiberg-Witten Floer homology.
• To establish a precise algebraic formula that computes the Casson invariant using the structure of monopole Floer homology.
• To extend the understanding of how Dehn surgery operations affect Seiberg-Witten invariants in the context of homology spheres.
• To provide a homological framework that connects classical invariants (Casson) with modern gauge-theoretic invariants (Seiberg-Witten).

Proposed method

• Utilizes the surgery exact triangle in monopole Floer homology to relate the Floer homology of a homology 3-sphere to that of a surgery on a knot.
• Applies duality and grading shift properties in monopole Floer homology to isolate the contribution of the Casson invariant.
• Employs the structure of the Seiberg-Witten Floer homology as a Z-graded module over the Novikov ring to extract the invariant.
• Relies on the fact that the Casson invariant equals the signed Euler characteristic of the Floer homology, adjusted by torsion corrections.
• Uses the fact that for a homology sphere, the Seiberg-Witten invariant is encoded in the Euler characteristic of the Floer homology.
• Applies the surgery formula to express the Casson invariant as a sum over spin^c structures, weighted by the graded dimensions of the Floer homology.

Experimental results

Research questions

• RQ1 How can the Casson invariant of a homology 3-sphere be algebraically reconstructed from its Seiberg-Witten Floer homology?
• RQ2 What is the precise relationship between the Casson invariant and the Euler characteristic of the monopole Floer homology?
• RQ3 To what extent does the surgery exact triangle in Floer homology encode the Casson invariant under Dehn surgery?
• RQ4 Can the conjectured formula by Kronheimer and Mrowka be proven using the structure of monopole Floer homology?
• RQ5 How do grading shifts and duality in Floer homology contribute to the computation of the Casson invariant?

Key findings

• The Casson invariant of a homology 3-sphere is equal to the signed Euler characteristic of its Seiberg-Witten Floer homology, up to a correction term arising from the torsion of the homology.
• The Dehn surgery formula expresses the Casson invariant as a sum over the graded dimensions of the Floer homology, weighted by the sign of the grading shift.
• The formula confirms the conjecture of Kronheimer and Mrowka by showing that the Casson invariant arises naturally from the structure of monopole Floer homology.
• The method establishes a direct link between classical topological invariants and modern gauge-theoretic invariants via exact triangle techniques.
• The result provides a homological interpretation of the Casson invariant as a count of generators in the Floer homology with signs determined by the spin^c structure and grading.
• The formula is invariant under orientation reversal and respects the duality properties of the Floer homology, ensuring consistency across surgery operations.