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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?

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1 Answer 1

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M is always aspherical, and hence a Eilenberg-Maclane space.

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.

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