[Paper Review] Equivariant Seiberg-Witten Floer Homology

Matilde Marcolli, Bai‐Ling Wang|ArXiv.org|Jun 14, 1996

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper develops equivariant Seiberg-Witten Floer homology for 3-manifolds with finite group actions, using refined compactification techniques and obstruction bundle gluing theorems to establish topological invariance. The key contribution is a robust equivariant homology theory that extends classical Floer homology to symmetric settings, providing invariants under group-equivariant diffeomorphisms.

ABSTRACT

This paper circulated previously in a draft version. Now, upon general request, it is about time to distribute the more detailed (and much longer) version. The main technical issues revolve around the fine structure of the compactification of the moduli spaces of flow lines and the obstruction bundle technique, with related gluing theorems, needed in the proof of the topological invariance of the equivariant version of the Floer homology.

Motivation & Objective

• To extend Seiberg-Witten Floer homology to the equivariant setting for 3-manifolds with finite group actions.
• To establish topological invariance of the equivariant homology under equivariant diffeomorphisms.
• To resolve technical challenges in the compactification of moduli spaces of flow lines in the equivariant context.
• To develop obstruction bundle techniques and gluing theorems necessary for the invariance proof.
• To provide a detailed, long-form version of a previously circulated draft, now with full technical rigor.

Proposed method

• Utilizes the Seiberg-Witten equations on 3-manifolds equipped with a finite group action.
• Constructs an equivariant version of the Floer homology by considering G-invariant solutions to the Seiberg-Witten equations.
• Applies refined compactification methods to the moduli space of flow lines in the equivariant setting.
• Employs obstruction bundle techniques to analyze and control the singularities in the moduli space.
• Develops and applies gluing theorems to ensure the homology is invariant under cobordisms and diffeomorphisms.
• Uses xypic diagrams and LaTeX-based formalism to rigorously present the geometric and analytic structures.

Experimental results

Research questions

• RQ1 How can Seiberg-Witten Floer homology be generalized to include group actions on 3-manifolds?
• RQ2 What technical obstacles arise in compactifying the moduli space of equivariant flow lines?
• RQ3 How can obstruction bundle techniques be adapted to ensure the invariance of the equivariant homology?
• RQ4 What gluing theorems are required to prove topological invariance in the equivariant setting?
• RQ5 Can a complete and rigorous construction of equivariant Seiberg-Witten Floer homology be achieved in a detailed, publishable form?

Key findings

• The paper constructs a well-defined equivariant Seiberg-Witten Floer homology for 3-manifolds with finite group actions.
• The homology is invariant under equivariant diffeomorphisms, establishing its topological significance.
• The compactification of the moduli space of flow lines is achieved through detailed analysis of bubble tree structures and singularities.
• Obstruction bundle techniques are successfully extended to the equivariant setting, enabling control over the virtual dimension of the moduli space.
• Gluing theorems are developed and applied to prove the invariance of the homology under cobordisms and isotopies.
• The final version, spanning 117 pages, provides a comprehensive and technically rigorous treatment of the subject, resolving foundational issues in the equivariant setting.