[Paper Review] Equivariant and Bott-type Seiberg-Witten Floer Homology: Part II

Rugang Ye|ArXiv.org|Jan 15, 1999

Geometric and Algebraic Topology27 references3 citations

TL;DR

This paper constructs equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, particularly rational homology spheres, using three equivalent versions—singular, de Rham, and Cartan—proving their diffeomorphism invariance. The work establishes a robust framework for equivariant Floer theory with applications in low-dimensional topology and gauge theory.

ABSTRACT

We construct equivariant and Bott-type Seiberg-Witten Floer homology and cohomology for 3-manifolds, in particular rational homology spheres, and prove their diffeomorphism invariance. We present several versions of the equivariant theory: the singular version, the de Rham version and the Cartan version, with the first playing the most important role. These versions are shown to be equivalent to each other. A few typos are removed.

Motivation & Objective

• To develop a comprehensive theory of equivariant Seiberg-Witten Floer homology for 3-manifolds, especially rational homology spheres.
• To introduce and compare three versions of the equivariant theory: singular, de Rham, and Cartan, establishing their equivalence.
• To extend the Seiberg-Witten Floer theory to include Bott-type Morse-Bott theory for non-degenerate critical manifolds.
• To prove the diffeomorphism invariance of the constructed homology and cohomology theories.
• To provide a foundation for future applications in gauge theory, geometric topology, and related areas of mathematical physics.

Proposed method

• The singular version of equivariant Seiberg-Witten Floer homology is constructed using singular homology with coefficients in a suitable local system associated to the circle action.
• The de Rham version is defined via differential forms on the space of connections modulo gauge, incorporating equivariant differential forms.
• The Cartan model is formulated using equivariant differential forms on the classifying space of the circle group, with values in the de Rham complex.
• Equivalence between the three versions is established through explicit chain homotopy equivalences and natural isomorphisms induced by the Chern-Weil construction.
• The theory is extended to the Bott-type setting by considering non-degenerate critical manifolds in the Chern-Simons functional, allowing for a Morse-Bott approach.
• Diffeomorphism invariance is proven by showing that the homology groups are independent of the choice of metric and perturbation, relying on continuation maps and gauge invariance.

Experimental results

Research questions

• RQ1 How can equivariant Seiberg-Witten Floer homology be consistently defined for 3-manifolds, particularly rational homology spheres?
• RQ2 What are the relationships between the singular, de Rham, and Cartan models of equivariant Floer homology, and are they equivalent?
• RQ3 Can the Bott-type Morse-Bott framework be incorporated into Seiberg-Witten Floer theory to handle non-isolated critical points?
• RQ4 How does the equivariant theory behave under diffeomorphisms of the underlying 3-manifold?
• RQ5 What are the implications of this theory for invariants in low-dimensional topology and gauge theory?

Key findings

• The singular, de Rham, and Cartan versions of equivariant Seiberg-Witten Floer homology are proven to be naturally isomorphic, establishing their equivalence.
• The Bott-type Seiberg-Witten Floer homology is constructed for 3-manifolds with non-degenerate critical manifolds in the Chern-Simons functional, generalizing the standard theory.
• The resulting homology and cohomology theories are invariant under diffeomorphisms of the 3-manifold, confirming their topological significance.
• The construction is valid for rational homology spheres, extending the applicability of equivariant Floer theory to a broad class of 3-manifolds.
• The theory provides a unified framework for studying gauge-theoretic invariants with symmetries, with potential applications in both topology and mathematical physics.