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I'm interested in proving the following claim:

Lemma: Let $M$ be an aspherical, closed 3-manifold which is not hyperbolic. Then, there exists a finite cover $M'$ of $M$ and an incompressible embedded torus $T$ in $M'$, such that $T$ is non-nullhomologous in $M'$.

I'm aware that if $M$ is Seifert fibered, then there exists a finite cover $M'$ of $M$ such that $M'$ is a circle bundle over a surface. I would like to say that we can take a further finite cover $M''$ which is a circle bundle over a surface $F$, where $F$ contains an embedded non-separating curve $\gamma$. Then the torus $T$ in $M''$ corresponding to the curve $\gamma$ is non-separating and thus non-nullhomologous.

If the above is correct, then it suffices to prove the Lemma in the case where the JSJ-decomposition of $M$ does not contain any Seifert fibered pieces. However, here I am stuck.

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  • $\begingroup$ Since the tori in the JSJ decomposition are by definition incompressible, it seems we can reduce to the case where the $3$-manifold is geometric, no? $\endgroup$
    – Nick L
    Aug 5, 2021 at 20:44
  • $\begingroup$ It’s possible, however, that the tori in the JSJ decomposition are all nullhomologous $\endgroup$
    – 24601
    Aug 5, 2021 at 21:11
  • $\begingroup$ Can you do something like Van Kampen? (as long as the pieces have non-trivial $\pi_{1}$) it seems it will work alot of the time (although I can't prove that). Is there some example where the JSJ tori are null-homologous? $\endgroup$
    – Nick L
    Aug 6, 2021 at 10:06
  • $\begingroup$ The tori are nullhomologous exactly when they are separating in the 3-manifold. How do you suggest I should apply Van Kampen in this situation? $\endgroup$
    – 24601
    Aug 7, 2021 at 9:31
  • $\begingroup$ I am now not so sure anymore sorry. Btw, It seems that there is a proof also for Sol $3$-manifolds. Take a finite cover w,hich is a $T^2$-bundle over $S^1$. Then, a $T^{2}$-fibre of the bundle is not seperating and incompressible. $\endgroup$
    – Nick L
    Aug 7, 2021 at 11:52

1 Answer 1

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Yes, this is true but the proof is somewhat long. You can extract it from the proof of the main result in

Hempel, John, Residual finiteness for 3-manifolds, Combinatorial group theory and topology, Sel. Pap. Conf., Alta/Utah 1984, Ann. Math. Stud. 111, 379-396 (1987). ZBL0772.57002.

Note that the paper operates under the "Haken" assumption, but it is only to ensure that Thurston's Geometrization Conjecture applies to the class of manifolds under consideration. Now, it's not a conjecture but a theorem (due to Perelman).

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