TITLE: When can a class in $H^1(M;\mathbb{Z})$ be represented by a fiber bundle over $S^1$
QUESTION [18 upvotes]: For a topological space M, It is known from homotopy theory that the elements of the first cohomology $H^1(M;\mathbb{Z})$ are in 1-1 correspondence with homotopy classes of maps $[M,S^1]$
In my case of interest M is a smooth manifold. Take $\alpha$ and take smooth $f\colon M \to S^1$ representing $\alpha$ under the above identification.
My question is, under what (sufficient / necessary / equiv) conditions $f$ gives a locally trivial fibration (Fiber bundle) of M over $S^1$?
I would prefer conditions in terms of homology / cohomology of M.
What comes to my mind is that, according to Ehrshmann theorem, since $S^1$ is compact,this map is a fibration iff surjective + submersion.
Surjectivity can be assured by picking nontrivial cohomology class.
As for submersion. I have no idea how to characterize.
Thanks

REPLY [3 votes]: Here is another way to approach this problem. Say $M$ is closed and orientable. The correspondance $H^1(M,\mathbb{Z})$ with $[M,S^1]$ works by pulling back the generator of $H^1(S^1,\mathbb{Z})$ along $f$ to get $\alpha$. 
Using Poincare duality and universal coefficient theorem (and forgetting about torsion) we can view $H^1(M,\mathbb{Z})$ as a lattice inside $H^1(M,\mathbb{R})$. By de Rham's theorem, $\alpha$ is represented by a closed $1$-form so that if we pick $f$ to be a smooth representative in the homotopy class, then we have $\alpha = f^*dx$ where $dx$ is the $1$-form on $S^1$ generating $H^1_{dR}(S^1)$. Then $f$ is a submersion if and only if $f^*dx$ is a nonvanishing $1$-form on $M$. So $\alpha$ corresponds to a fiber bundle if and only if, after tensoring with $\mathbb{R}$, it can be represented by a nonvanishing closed $1$-form.
In fact one can prove something slightly stronger as is done in this paper. A closed orientable manifold $M$ is fibered over $S^1$ if and only if it has a nonvanishing closed $1$-form and in this case the fibers are the leaves of the foliation generated by this $1$-form. Furthermore, if we have a nonvanishing closed $1$-form $\omega$ that is a nonero real multiple of $f^*dx$, then the fiber bundle map induced by $\omega$ is isotopic to $f$.