What's the "easiest" closed 3-manifold with a nonabelian fundamental group?

Question

I'm looking for some easy compact, oriented 3-manifolds without boundary that have a nonabelian fundamental group.

It needn't be perfect. "Easy" means that it has an easy Heegard diagram, say, one that could be memorised and drawn by heart. The Poincaré Homology Sphere does not have an easy Heegard diagram, as far as I know, see http://www.its.caltech.edu/~wjiajun/compprog/hfhat/sigma235.html.

All the examples of 3-manifolds I know and I can find are either not closed ($\mathbb{R}^3$ with two $D^2 \times \mathbb{R}$ taken out, $D^2$ being a 2-disk) or have an abelian fundamental group ($S^3$, $\mathbb{RP}^3$ etc.) or are horribly complicated (Poincaré Homology Sphere).

Answer 1

The product of a circle with a closed genus 2 surface $X$ has fundamental group $\mathbb{Z}\times\pi_1(X)$, where $\pi_1(X)$ is nonabelian with presentation $$\langle a,b,c,d | aba^{-1}b^{-1},cdc^{-1}d^{-1}\rangle$$

Edit: Or just take a genus 2 handlebody. This has fundamental group isomorphic to the free group on 2 generators. If I understand Heegaard diagrams correctly, its heegaard diagram should be trivial...

Answer 2

Look at pushouts. Any null homologous essential surface in a 3-manifold gives you a decomposition as an amalgamated product $\pi_1(M)= \pi_1(M_1) *_{\pi_1(S)}\pi_1(M_2)$.

In particular you get connected sums $M_1\# M_2$ gives you free products on fundamental group since $S^2$ is simply connected: $\pi_1(M_1\# M_2) = \pi_1(M_1)*\pi_1(M_2)$.

So if you ask for the simplest example you might take your favorite simple non-simply connected closed 3-manifold and take the connected sum with itself.