Rank three instantons, representations and sutures

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 $S^1\times \Sigma_g$. 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.