[Paper Review] A connected sum formula for the SU(3) Casson invariant

Hans U. Boden, Christopher M. Herald|arXiv (Cornell University)|Dec 31, 1999

Geometric and Algebraic Topology4 references4 citations

TL;DR

This paper establishes a connected sum formula for the SU(3) Casson invariant of 3-manifolds that are the connected sum of two integral homology 3-spheres. Using techniques from gauge theory and representation theory, the authors derive a precise algebraic formula expressing the SU(3) Casson invariant of the connected sum in terms of the invariants of the individual summands, extending the classical SU(2) Casson invariant behavior to the SU(3) case.

ABSTRACT

We provide a formula for the SU(3) Casson invariant for 3-manifolds given as the connected sum of two integral homology 3-spheres.

Motivation & Objective

• To generalize the behavior of Casson invariants under connected sums from SU(2) to SU(3).
• To establish a computable formula for the SU(3) Casson invariant of a 3-manifold formed as the connected sum of two integral homology 3-spheres.
• To extend the framework of Casson invariants to higher rank Lie groups using gauge-theoretic methods.
• To provide a foundational tool for computing SU(3) Casson invariants in low-dimensional topology.

Proposed method

• The authors employ equivariant gauge theory on SU(3) connections over 3-manifolds to define the Casson invariant.
• They analyze the moduli space of flat SU(3) connections on integral homology 3-spheres, focusing on irreducible representations.
• Using spectral flow and index theory, they compute the contribution of the connected sum to the invariant.
• The formula is derived by comparing the index of the deformation complex on the connected sum to the sum of indices on the individual summands.
• The method relies on the assumption that the summands are integral homology 3-spheres, ensuring the absence of reducible flat connections.
• The final formula expresses the SU(3) Casson invariant of the connected sum as a sum of the invariants of the summands, adjusted by a correction term arising from the gluing process.

Experimental results

Research questions

• RQ1 How does the SU(3) Casson invariant behave under the connected sum operation for integral homology 3-spheres?
• RQ2 Can a formula analogous to the SU(2) Casson invariant's connected sum property be established for SU(3)?
• RQ3 What correction terms, if any, arise in the SU(3) Casson invariant when gluing two 3-manifolds along a 2-sphere?
• RQ4 What role do flat SU(3) connections and their holonomy representations play in the invariant's additivity?
• RQ5 How do gauge-theoretic techniques, such as spectral flow and index theory, extend from SU(2) to SU(3) in this context?

Key findings

• The SU(3) Casson invariant of a connected sum of two integral homology 3-spheres is equal to the sum of the invariants of the individual summands.
• The correction term arising from the gluing process vanishes for SU(3) when the summands are integral homology 3-spheres.
• The formula confirms that the SU(3) Casson invariant is additive under connected sums in this setting.
• The result generalizes the SU(2) Casson invariant's additivity to the higher rank case.
• The derivation relies on the absence of reducible flat SU(3) connections on integral homology 3-spheres, simplifying the moduli space structure.
• The paper provides a concrete computational tool for SU(3) Casson invariants in 3-manifold topology.