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I asked this question on MSE 9 days ago and it got a very helpful comment from Eric Towers providing the Palais Stewart reference, but no answers. So I'm crossposting it here.

Let $$ T^n \to M \to T^m $$ be a principal torus bundle over a torus. Then $ M $ is a solvmanifold, even a nilmanifold (in fact $ M $ is the total space of a principal torus bundle over a torus if and only if it is a compact nilmanifold for a 2 step nilpotent Lie group, this is theorem 3 of [Palais, Stewart, TORUS BUNDLES OVER A TORUS]).

What if the bundle is not necessarily principal? Is every torus bundle over a torus a solvmanifold? In other words, if we have a fiber bundle $$ T^n \to M \to T^m $$ then can we conclude that the total space $ M $ is a solvmanifold?

EDIT:

The answer, comments and references from Igor Belegradek prove that something much stronger is true: A manifold $ M $ is the total space of a bundle $$ N \to M \to T^n $$ where $ N $ is a compact nilmanifold and $ T^n $ is a torus if and only if $ M $ is homeomorphic to a compact solvmanifold.

The smooth case is also addressed. Corollary 2.3 of https://arxiv.org/pdf/1307.3223.pdf states that "If $ h: T^n → T^n $ , $ n ≥ 6 $, is a diffeomorphism given by Proposition 2.1 then the mapping torus $ M_h $ is a fake torus." By fake torus the authors mean $ M_h $ is homeomorphic to a torus but not PL homeomorphic. Since $ M_h $ is not PL homeomorphic to a torus it is also not diffeomorphic to a torus. Thus already in dimension 7 we have an exotic homotopy torus $ M_h $ which is the total space of a torus bundle over the torus $$ T^6 \to M_h \to S^1 $$ but which is not diffeomorphic to any solvmanifold. So in every dimension $ d \geq 7 $ there are smooth torus bundles over a torus (both base and fiber with the standard smooth structure) that are not diffeomorphic to any solvmanifold.

For $ d \leq 3 $ we have by Moise's theorem that there are no exotic smooth structures. So all torus bundles over a torus with total space of dimension $ d \leq 3 $ are diffeomorphic to a solvmanifold.

For $ d=4 $ the existence of exotic 4 tori (or any exotic closed aspherical manifold) seems to be an open problem see

Do there exist exotic 4-tori?

In that same question it is pointed out that there are exotic tori in $ d=5,6 $ but it is not clear that these can be constructed as smooth fiber bundles with base and fiber standard tori.

That said, for the $ d=4 $ case a torus bundle over a torus has base and fiber with dimension $ \leq 3 $ so there shouldn't be any exotic gluing maps and it seems like the total space of such a bundle should just always have the standard smooth structure.

For $ d=5,6 $ I have no idea what kind of smooth structures might or might not be possible on torus bundles over the torus.

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    $\begingroup$ Smooth $T^n$ bundles over $T^m$ and classified by homotopy classed of maps from $T^m$ into $BDiff(T^n)$, and $Diff(T^n)$ is fairly complicated. In some dimensions its identity component is not homotopy equivalent to a finite CW complex (theorem B in "The non-finite homotopy type of some diffeomorphism groups" by Antonelli- Burghelea-Kahn, sciencedirect.com/science/article/pii/…) and $\pi_0(Diff(T^n))$ is often not $GL(n,\mathbb Z)$, see theorem 2.5 in "Parametrized surgery and isotopy" by Hsiang-Sharpe, msp.org/pjm/1976/67-2/pjm-v67-n2-p09-s.pdf. $\endgroup$
    – Igor Belegradek
    Commented Feb 20, 2022 at 16:45
  • $\begingroup$ I am worried about your claim "in every dimension $d≥7$ there are smooth torus bundles over a torus not diffeomorphic to solvmanifolds". I think Hitchin uses homotopy spheres that first show up in dimensions $8$ and $9$, and then I guess one can take products with tori. I haven't thought this through but I don't see how $d$ could be $7$ if one uses Hitchin spheres. I also think one should be able to see what happens if $d=4$, and I expect that all 4-dimensional torus bundles over tori should be diffeomorphic to solvmanifolds. $\endgroup$
    – Igor Belegradek
    Commented Feb 21, 2022 at 21:19
  • $\begingroup$ I don't know very much about this topic. I know that there are homotopy 7 spheres which have an exotic smooth structure. Are you saying that only certain types of exotic homotopy spheres are appropriate for this construction and that exotic spheres of that particular type do not arise until dimension 8? $\endgroup$
    – Ian Gershon Teixeira
    Commented Feb 22, 2022 at 12:35
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    $\begingroup$ What you claim (about existence of exotic tori bundle over tori in each dimension $\ge 7$) might still be true but the reference I gave to Hajduk-Tralle's paper involved Hitchin homotopy spheres and those do not exist in dimension $7$. In any case you referred to me for something I did not say, and I wanted to set things straight. What I am reasonably sure of at this moment is exotic tori bundle over tori in some dimensions that start at $8$. However, section 2 of arxiv.org/pdf/1307.3223.pdf may give what you claimed but I haven't looked at it closely. $\endgroup$
    – Igor Belegradek
    Commented Feb 22, 2022 at 12:49
  • $\begingroup$ Thanks for the new reference! I've edited my post to better reflect the truth. $\endgroup$
    – Ian Gershon Teixeira
    Commented Feb 22, 2022 at 22:29

1 Answer 1

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A group $G$ is isomorphic to the fundamental groups of a compact solvmanifold if and only if it fits into the short exact sequence $1\to N\to G\to\mathbb Z^n\to 1$ where $N$ is a finitely generated torsion-free nilpotent group. This is stated on p.253 and explained is chapter III of Auslander's An exposition of the structure of solvmanifolds. Part I: Algebraic theory. In particular, every torus bundle over a torus is homotopy equivalent to a solvmanifold.

You may also be interested in Theorem 3 of Wilking's paper Rigidity of group actions on solvable Lie groups which gives an analogous result for infrasolvmanifolds.

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  • $\begingroup$ That's a beautiful result up to homotopy equivalence! I guess to complete the story I would want to know if every torus bundle over a torus is actually homeomorphic to a solvmanifold? (I know diffeomorphism is too much to ask for since there are exotic smooth tori in dimension $ n\geq 5 $ which cannot be diffeomorphic to a solvmanifold.) Are there perhaps any results in low dimensions? For example, in dimension 3 is it true that every torus bundle over a torus is diffeomorphic or at least homeomorphic to a solvmanifold? $\endgroup$
    – Ian Gershon Teixeira
    Commented Feb 20, 2022 at 17:49
  • $\begingroup$ Yes, one can improve the statement from homotopy equivalence to homeomorphism because Borel conjecture is true for closed aspherical manifolds with virtually solvable fundamental group , which covers the case of dimension $>4$ and in dimension $4$ one uses that virtually solvable group are good, see e.g. theorem 1-2 in him.uni-bonn.de/lueck/data/indam.pdf and p.3 in him.uni-bonn.de/lueck/data/indam.pdf. It might be possible to further improve to a diffeomorphism, but I don't know how to do that. Exotic tori are not counterexamples, I think. $\endgroup$
    – Igor Belegradek
    Commented Feb 20, 2022 at 21:17
  • $\begingroup$ The first reference is great thanks so much. I think you left out the second reference and accidentally put in the first reference again. Here is why I think exotic tori are counter examples: It is a theorem of Auslander that compact solvmanifolds are determined up to diffeomorphism by their fundamental group. The torus with standard smooth structure is a solvmanifold. So any homotopy torus diffeomorphic to a compact solvmanifold must be diffeomorphic to the torus with standard smooth structure. $\endgroup$
    – Ian Gershon Teixeira
    Commented Feb 20, 2022 at 23:28
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    $\begingroup$ Sorry, here's the second reference arxiv.org/pdf/2006.05209.pdf. Regarding your other comment, the point is that I don't know if any exotic torus is a smooth torus bundle over a torus. It may be that any smooth torus bundle over a torus is actually diffeomorphic to a solvmanifold. $\endgroup$
    – Igor Belegradek
    Commented Feb 21, 2022 at 1:59
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    $\begingroup$ Actually it is explained on p.1 of arxiv.org/pdf/math/0311412.pdf that there is an exotic torus that is the total space of a fiber bundle whose base and fiber are standard tori. $\endgroup$
    – Igor Belegradek
    Commented Feb 21, 2022 at 3:40

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