# Irreducible 3-manifold with boundary of genus greater than 1

Let $$M$$ be an irreducible 3-manifold with incompressible boundary of genus > 1.

When is $$M$$ homotopy equivalent to an Eilenberg-MacLane space? Or it is never true?

This is because $$\pi_1(\partial M)$$ embeds to $$\pi_1(M)$$, by the incompressibility condition. If the genus of the boundary is no less than 1, then $$M$$ has infinite fundamental group. This forces the universal cover $$\tilde M$$ to be a non-compact 3-manifold, so $$H_i(\tilde M)=0,\ i\geqslant 3$$.
By sphere theorem, $$M$$ is irreducible implies that $$\pi_2(\tilde M)=\pi_2(M)=\{e\}$$. Note that $$\pi_1(\tilde M)=\{e\}$$, then by Hurewicz theorem we have $$H_i(\tilde M)=0,\ i=1,2.$$ Then $$\tilde M$$ has trivial integral homology and is simply connected, so $$\tilde M$$ is contractible by Whitehead theorem.