# $SU(n)$ character variety of integral homology spheres

Suppose that $$Y$$ is an integral homology $$3$$-sphere. Is it true that, for any $$n \in \mathbb{N}$$, there is always an irreducible representation $$\pi_{1}(Y) \rightarrow SU(n)$$?

I'm also happy to see the answer when the Lie group is replaced by $$SL(n, \mathbb{C})$$ or $$PSL(n, \mathbb{C})$$, but keeping irreducible unchanged.

Edit: In this paper, Zentner proved the statement positively for $$SL(2, \mathbb{C})$$. It might be true that there is a threshold $$n$$ such that there is no more $$SU(N)$$ irreducible representations for $$N \geq n$$. The question was stated in this way because constructing counter-examples to the vanishing statement might be easier.

• The Poincaré homology sphere has fundamental group the binary icosahedral group, which is finite and therefore admits only finitely many irreducible representations. Jul 5 '20 at 13:16
• Yes, my motivation of asking this question is to see if this failure holds in much more generality. Jul 5 '20 at 16:56
• So you are asking: for all $n \in \mathbb N$, is there an integral homology 3-sphere $Y$ with an irreducible $\pi_1(Y) \to SU(n)$? Jul 5 '20 at 17:29
• No, I am asking if $Y$ is fixed, can you find such representations. As you pointed out this is not true if Y is the Poincar\'e homology sphere, the question is about whether this failure holds for all $Y$. Jul 5 '20 at 18:11
• It should be true that for a given closed 3-manifold with infinite fundamental group, there is an irreducible $SU(N)$ representation for $N$ sufficiently large. But the case of homology spheres and $n=2$ is a well-known open question, so I think your question as stated is probably difficult. Jul 5 '20 at 20:15