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Rank three instantons, representations and sutures

Aliakbar Daemi, Nobuo Iida, Christopher Scaduto

Abstract

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 $S^1\times Σ_g$. 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.

Rank three instantons, representations and sutures

Abstract

We show that the knot group of any knot in any integer homology sphere admits a non-abelian representation into such that meridians are mapped to matrices whose eigenvalues are the three distinct third roots of unity. This answers the case of a question posed by Xie and the first author. We also characterize when a -bundle admits a flat connection. The key ingredient in the proofs is a study of the ring structure of instanton Floer homology of . 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 instanton Floer homology of sutured manifolds, and then obtain the mentioned topological applications. Along the way, we prove a structure theorem for Donaldson invariants, which is the counterpart of Kronheimer and Mrowka's structure theorem for Donaldson invariants. We also prove a non-vanishing theorem for the Donaldson invariants of symplectic manifolds.
Paper Structure (15 sections, 40 theorems, 317 equations, 1 figure)

This paper contains 15 sections, 40 theorems, 317 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and general strategy
  3. Mumford relations and their duals
  4. The $N=3$ undeformed simple type quotient
  5. Sutured instanton homology
  6. The Structure Theorem
  7. Fukaya--Floer homology of $S^1\times \Sigma_g$
  8. Proof of the structure theorem
  9. A non-vanishing theorem for symplectic 4-manifolds
  10. $U(N)$ framed instanton homology
  11. Definition
  12. Euler characteristic
  13. A product formula for connected sums
  14. A decomposition result for cobordism maps
  15. $N=3$ knot homology and the Alexander polynomial

Key Result

Theorem 1

If $K$ is a non-trivial knot in an integer homology $3$-sphere $Y$, then there exists a homomorphism $\phi:\pi_1(Y\setminus K)\to SU(3)$ with non-abelian image, such that

Figures (1)

  • Figure 1: The curves $C_i$ for $1\leq i \leq 2g+1$ on a surface of genus $g$.

Theorems & Definitions (89)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Theorem 6
  • Remark 2
  • Lemma 1
  • proof
  • ...and 79 more