It is apparently a result of F. González-Acuña that all closed orientable 3-manifolds contain a fibered knot. (I am not sure exactly where to find a published proof of this result and as an aside I would be interested in hearing about any proofs/references that anyone knows.)


I am wondering to what extent this is true in higher dimensions and specifically in dimension 4. In particular, is there always a fibered 2-sphere in a closed orientable 4-manifold? If not, what about arbitrary genus surfaces?



Existence of fibered surfaces in arbitrary 4-manifolds?
I suppose that, for a $n$-manifold $M$, containing a "fibered codimension-2 manifold" $N\subset M$ means that $N$ has trivial normal neighbourhood $\nu N = N\times D^2$ and its complement $M \setminus \nu N$ fibers over $S^1$ with a fibration that extends the projection $\partial (\nu N) = N \times S^1 \to S^1$ of its boundary. The fiber of the fibration is some $(n-1)$-manifold $Y$ with boundary $N$. 


Any such manifold $M$ is obtained from a $(n-1)$-manifold with boundary $Y$ and a self-diffeomorphism $\varphi$ of it (that restricts to the identity on the boundary). You use $\varphi$ as the monodromy to construct a fibration over $S^1$ with fiber $Y$, and then cap the boundary component by gluing $\partial Y \times D^2$. Following Gordon, when $\varphi$ is trivial, we may say that $M$ is obtained by  *spinning*  $Y$. In this case we get $\pi\_1(M) = \pi\_1(Y)$.


The first thing to do is to calculate some topological invariants of such a manifold $M$ obtained in this way. We get $\chi(M) = \chi(N)$. So if a four-manifold $M$ contains a fibred 2-sphere it must have $\chi(M)=2$. If it contains a more general fibred connected surface $N$, we get $\chi(M) = \chi(N) \leq 2$. I would also bet that $M$ is even and has zero signature (I haven't checked this). One could look at the possible fundamental groups $\pi\_1(M)$ one could get. Geometrisation of 3-manifolds gives some information on the possible $Y$ and the possible $\varphi$.


