Are rational homology 3-spheres atoroidal? I can't seem to pin down whether rational homology 3-spheres are atoroidal. For a 3-manifold to be atoroidal then there must be no $\pi_{1}$-injective tori. That is, there is no injection $\mathbb{Z}\oplus\mathbb{Z}\hookrightarrow \pi_{1}(M)$.
Of course, we cannot simply abelianize since there are 3-manifolds where there is an injection $\mathbb{Z}\oplus\mathbb{Z}\hookrightarrow \pi_{1}(M)$, but $\mathbb{Z}\oplus\mathbb{Z}\to H_{1}(M)$ is not injective. (At least, I'm not certain of this, but I could believe it. For example, the group $G = \langle a,b | a^{2} = b^{2} = 1\rangle$ has a subgroup generated by $ab$ that is isomorphic to $\mathbb{Z}$, but the abelianization $Ab(G)$ of $G$ has no injection $\mathbb{Z}\to Ab(G)\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$).
Moreover, every nontrivial class in $H_{2}(M;\mathbb{Z})$ can be realized by a closed essential orientable surface. So if $M$ is a rational homology sphere, this doesn't rule out the fact that $M$ could have closed orientable essential tori. However, if $M$ contains an embedded incompressible torus $T$, then wouldn't the embedding $i:T\to M$ induce an injection $i_{*}:H_{1}(T)\to H_{1}(M)$?
I could be horribly on the wrong track, so any help would be appreciated!
 A: No, this is quite false. Even integer homology 3-spheres can contain incompressible tori. Let $K$ be a (smooth or polygonal) knot in $S^3$, the exterior $E(K)$ of $K$ is the complement in $S^3$ to an open tubular neighborhood of $K$. Thus, $\partial E(K)$ is homeomorphic to the 2-dimensional torus $T^2$. Furthermore, $H_*(E(K))\cong H_*(S^1)$ and the homomorphism induced by the inclusion $H_1(\partial E(K))\to H_1(E(K))$ is surjective. In particular, one of the simple loops on the boundary torus generates $H_1(E(K))$. (Proving these statements is a nice exercise in algebraic topology.) A less trivial fact is that if $K$ is a nontrivial knot (i.e. is not an unknot) then $\partial E(K)$ is incompressible in $E(K)$. One can derive this, for instance, from the Loop Theorem.
Now,  let $M_1, M_2$ be exteriors of some nontrivial knots in $S^3$.
Next, glue $M_1, M_2$ by a homeomorphism $h$ of their boundaries such that $h$ sends a loop generating  $H_1(M_1)$ to a null-homologous loop on the boundary of $M_2$ and vice-versa. The result will be a closed 3-manifold $M$ which is integer homology sphere containing an incompressible torus, namely, the common image  of $\partial M_1, \partial M_2$ in $M$. (The first claim is an exercise in application of the Mayer-Vietoris sequence; the second is    an exercise in application of the Seifert- Van Kampen Theorem.)
