TBA

We will discuss the problem of finding hyperbolic manifolds fibring over the circle; and show a method to construct and analyse maps from particular hyperbolic manifolds to S^1, which relies on Bestvina-Brady Morse theory. 
This technique can be used to build and detect fibrations, algebraic fibrations, and Morse functions with minimal number of critical points, which are interesting in the even dimensional case. 
After an introduction to the problem, and presentation of the main results, we will use the remaining time to focus on some easy 3-dimensional examples, in order to explicitly show the construction at work.
 

I will discuss three different constructions of smooth tori in S^4 whose complements have fundamental group Z: turned 1-twist-spun tori due to Boyle, the union of a ribbon disc with a genus one Seifert surface constructed by Cochran and Davis, and certain tori with four critical points. They are all topologically unknotted, but it is not known whether they are smoothly standard, except for tori with four critical points whose middle level set is a split link. The branched double cover of S^4 along any of these surfaces is a potentially exotic copy of S^2 x S^2, though, in the case of Boyle's example, it cannot be distinguished from the standard S^2 x S^2 using Seiberg-Witten invariants. This is joint work with Mark Powell.