Is there a nice reference for the classification of closed 3-manifolds with solvable (nilpotent, abelian, etc.) fundamental group, assuming the Geometrization Conjecture?

  • $\begingroup$ To follow up, I was hoping for a citable source with a theorem of the form: If $\pi_1(M)$ is abelian, $M$ is homeomorphic to one of the following ... If it is nilpotent, but not abelian... If it is solvable, but not nilpotent... The Evans-Moser paper comes pretty close, but one still has to argue that "equivalent" in that paper is the same as homeomorphic, due to the Poincaré Conjecture. $\endgroup$ – Andy Hammerlindl Jan 18 '12 at 16:21

Peter Scott, The geometries of 3-manifolds, Bull. London Math. Soc., 15 (1983), 401-487.

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    $\begingroup$ While Scott's survey sure is a nice reference, does it really give a full classification of closed 3-manifolds with solvable fundamental group? I do not think it stated there. After brief search I found the classification of compact 3-manifolds with amenable fundamental group in section 4 of math.fsu.edu/~aluffi/archive/paper409.pdf $\endgroup$ – Igor Belegradek Jan 17 '12 at 21:30
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    $\begingroup$ While it's not explicitly stated, I think you can extract it without too much fuss. The manifold will either be Seifert fibered or modeled on Sol. In the first case, the classification of Seifert fibered spaces kicks in, and in the second, it's a torus or Klein bottle bundle and the homotopy class of the monodromy determines the manifold. $\endgroup$ – Autumn Kent Jan 17 '12 at 22:47
  • $\begingroup$ OK, I guess you have to be a little careful since manifolds can admit different Seifert fiberings or bona fide fiberings. $\endgroup$ – Autumn Kent Jan 17 '12 at 22:53
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    $\begingroup$ It's probably not a hard exercise either to find the constant curvature manifolds (spherical, Euclidean) with solvable fundamental group. $\endgroup$ – Ian Agol Jan 18 '12 at 0:20

Here are some references that may help answer your question:

  • Charles B. Thomas, Nilpotent groups and compact $3$-manifolds, Proc. Cambridge Philos. Soc. 64 (1968), 303-306; MR0233359.
  • Benny Evans and Louise Moser, Solvable fundamental groups of compact $3$-manifolds, Trans. Amer. Math. Soc. 168 (1972), 189–210; MR0301742.
  • Peter Teichner, Maximal nilpotent quotients of $3$-manifold groups, Math. Res. Lett. 4 (1997), no. 2-3, 283-293; MR1453060.

    A little nugget (due to John Milnor): among the Brieskorn manifolds $\Sigma(p,q,r)$, the only nilmanifolds are $\Sigma(2,3,6)$, $\Sigma(2,4,4)$, and $\Sigma(3,3,3)$, which are circle bundles over the torus with Euler number $1$, $2$, and $3$, respectively.

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    Although I am late to answer this question, I wanted to put in a sales pitch for the flow charts at the end of Thurston's book (Figures 4.22 and 4.23). They do a great job for dealing with your questions for oriented manifolds.

    In the first chart, Thurston uses "almost" to mean virtually, i.e. has a finite index subgroup with some property. Considering compact manifolds with abelian fundamental group, we have the following classification. (The references are to Thurston's book although older references exist for nearly all of these statements.) If $\pi_1(M)$ is finite, then $M$ is a lens space with finite cyclic fundamental group (see Theorem 4.4.14). If $\pi_1(M)$ is not finite and abelian, then $\pi_1(M)= Z, Z\times Z\times Z$ . The first case implies that $M\cong S^1\times S^2$ (see Exercise 4.7.1) and the second implies that $M \cong T^3$ (see Theorem 4.3.4).

    As Richard Kent points out the paper of Peter Scott is a wonderful reference and these last two facts are discussed in greater detail there.

    Finally to finish our classification for oriented manifolds, we also have to worry about the cusped manifolds. In general, the second chart has this information. The one caveat is that the unknot and Hopf link complements ($S^1\times D^2$ and $T^2 \times I$) might not neatly fit on this list, because they do not admit a cofinite $H^3$, $\widetilde{PSL(2,R)}$ or $H^2 \times R$ structures. However, unknot and Hopf link complements should be added to our list of manifolds having abelian fundamental groups as well since their fundamental groups are $Z$ and $Z\times Z$, respectively. I believe they are not considered in this chart because they would not show up in a decomposition prescribed by the Geometrization. For example, if you attach a Hopf link complement to the boundary of geometric piece via a boundary identification the resulting manifold remains geometric.

    The non-orientable case, includes $S^1 \times P^2$, which has fundamental group $Z \times Z/2Z$ (see Epstein's paper Theorem 9.1).


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