# Foliations by circles on the 3-torus

Let $T=S^1\times S^1 \times S^1$ be the 3-torus. Let us denote by $\alpha$ the isotopy class of the loop $S^1\times pt\times pt$ and let $\mathcal F_\alpha$ be the set of all smooth oriented foliations $F$ of $T$ by circles such that every leaf of $F$ lies in the class $\alpha$ (in particular $F$ yields a smooth principal $S^1$-bundle over the 2-torus having $T$ as total space).

Question: Is $\mathcal F_\alpha$ connected?

• if the $S^1$ fibers tangle in any way, then I suspect they are not homotopic to the "trivial" foliation, $S^1 \times pt$. Is that right? – john mangual Mar 9 '17 at 16:19
• A: Yes. Such foliations are Seifert fiberings, Using things like Alexander's theorem you can classify the the Seifert fiberings of manifolds that admit them, even up to isotopy. For example, take a look in Hatcher's 3-manifolds notes. – Ryan Budney Mar 9 '17 at 16:31
• The fact that circle foliations are Seifert fibrations (for arbitrary 3-manifold, even a noncompact one, and even without smoothness assumption) is due to D.B.A.Epstein "Periodic flows on 3-manifolds" Annals of Math, 1972. – Misha Mar 9 '17 at 21:16
• @johnmangual: I edited the question, so that now $\alpha$ is the isotopy class of the loop and not the homotopy class (maybe they are different and this could answer your point). – Gabriele Benedetti Mar 10 '17 at 19:16
• @RyanBudney: Thanks a lot for the reference. I had a look at Hatcher's notes. For him $T^3$ is the exceptional case $M_1$ on p.37, for which he does not seem to provide the isotopy classification (but maybe I am wrong). I also found Theorem 5.2 (originally due to Waldhausen) in these notes by Jankins and Neumann math.columbia.edu/~neumann/preprints/… which asserts that except some cases ($T^3$ is case iii) two homeomorphic Seifert fiberings are actually isotopic. – Gabriele Benedetti Mar 10 '17 at 19:33

If I understood correctly it is enough to show that there exists a surface of section isotopic to $pt \times S^1 \times S^1$. After isotopy, you can assume that that surface is indeed the section and all circles of the foliation have the same length (see the paper D.B.A.Epstein "Periodic flows on 3-manifolds" Annals of Math, 1972. cited by Misha above). This allows to construct the desired diffeotopy to the canonical one.