2024/02/16 by Aliakbar Daemi, Daemi, Aliakbar, Nobuo Iida +3 · 1 citation
Mathematics · #57R58 57M05 57K18 14H60 #FOS: Mathematics #Geometric Topology (math.GT) #History and Theory of Mathematics
paper · pdf · doi:10.48550/arxiv.2402.10448
openalex publication_date 2024/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that the knot group of any knot in any integer homology sphere admits a non-abelian representation into SU(3) such that meridians are mapped to matrices whose eigenvalues are the three distinct third roots of unity. This answers the N=3 case of a question posed by Xie and the first author. We also characterize when a PU(3)-bundle admits a flat connection. The key ingredient in the proofs is a study of the ring structure of U(3) instanton Floer homology of S1× Σg. In an earlier paper, Xie and the first author stated the so-called eigenvalue conjecture about this ring, and in this paper we partially resolve this conjecture. This allows us to establish a surface decomposition theorem for U(3) instanton Floer homology of sutured manifolds, and then obtain the mentioned topological applications. Along the way, we prove a structure theorem for U(3) Donaldson invariants, which is the counterpart of Kronheimer and Mrowka's structure theorem for U(2) Donaldson invariants. We also prove a non-vanishing theorem for the U(3) Donaldson invariants of symplectic manifolds.