There are many non-homeomorphic 3-manifolds with isomorphic fundamental groups, for example the lens spaces $L(p,q_1)$ and $L(p,q_2)$ with $q_1 \ne \pm q_2^{\pm 1}\mod p$. Also, Seifert fibered spaces over the disk with two exceptional fibers can have isomorphic fundamental groups. A third type of example can be constructed by using the connect sum of knot of a chiral knot $K$, $S^3-(K\#K)$ and $S^3-(K\#-K)$ (think of the square and granny knots). Even more examples can be constructed by connect summing the examples in this paragraph (in the sense of 3-manifolds as opposed to knots).

However, there are also non-orientable and orientable manifolds with isomorphic fundamental groups, which motivates my question:

Is there a list of all non-homeomorphic pairs of 3-manifolds (orientable and non-orientable) with isomorphic fundamental groups?

Given, the likely complexities of reducible manifolds and graph manifolds, it would be fine if it was only a list of geometric 3-manifolds.

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    $\begingroup$ Aschenbrenner, Friedl and I discuss this question at the beginning of our survey '3-manifold groups'. In the closed case, non-homeomorphic pairs only arise as lens spaces or connect sums, but the case with boundary is more complicated. $\endgroup$ – HJRW Apr 29 '15 at 11:05
  • $\begingroup$ Thanks Henry. Isn't assumed throughout your paper that you guys discuss only oriented 3-manifolds? I was really hoping to find a reference that deals with the non-orientable case as well. I fully understand why one would like to avoid the pathologies of non-orientable manifolds especially when considering virtual properties. However, I wondering if anyone has a nice approach to 3-manifolds with the same fundamental group regardless of orientablity. $\endgroup$ – Neil Hoffman Apr 29 '15 at 12:07
  • $\begingroup$ Sorry, Neil, I missed your emphasis on the non-orientable case. Yes, we don't discuss that. Indeed, I think it's not even clear that geometrization has been worked our properly in the non-orientable case, so you may want to assume that. $\endgroup$ – HJRW Apr 29 '15 at 12:12

(Edit: Everything what follows is about closed and orientable $3$-manifolds.)

Non-spherical geometric $3$-manifolds are determined by their fundamental group. This is proved in Peter Scott's paper "There are no fake Seifert fiber spaces with infinite $\pi_1$" http://www.jstor.org/stable/2006970 (He does the case of Seifert fibrations. For hyperbolic $3$-manifolds the uniqueness of course follows from Mostow rigidity.)

For irreducible manifolds with nontrivial JSJ-decomposition one can apply Waldhausen's rigidity theorem (because these manifolds are necessarily Haken) to conclude that the are also determined by their fundamental groups.

For reducible $3$-manifolds things are a bit more complicated because one can choose orientations of the prime components separately. Section 2 of the survey paper of Aschenbrenner-Friedl-Wilton http://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/3-manifold-groups-final-version-031115 explains why there are only three ways to get 3-manifolds with isomorphic fundamental groups.These three ways are

  • the examples of lens spaces mentioned in the question.

  • manifolds of the form $M\sharp N$ and $M\sharp \overline{N}$, where $N$ and $\overline{N}$ are the same manifold with opposite orientations

  • examples constructed from the above by taking connected sums.

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  • $\begingroup$ I think you might want to be a little careful citing Scott. He also requires that the manifolds are closed, right? There are examples of non-homomorphic Seifert fibered spaces over the disk with two exceptional fibers that are non-homeomorphic. D^2(3,1)(3,1) and D^2(3,1)(3,2) form a pair of these type of examples. One can see they are not homeomorphic by performing surgery along the peripheral curve that bounds an annulus, which results in L(3,1)#L(3,1) and L(3,1)#L(3,2). $\endgroup$ – Neil Hoffman Apr 29 '15 at 12:03
  • $\begingroup$ Yes, I was talking about closed manifolds. In section 2.2 of Aschenbrenner-Friedl-Wilton you find a discussion of $3$-manifolds with incompressible boundary. $\endgroup$ – ThiKu Apr 29 '15 at 12:05

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