@DaveBenson Thanks. I only know Schubert calculus for the usual complex or real Grassmannians. Presumably $BSU(n)$ is modeled on some Grassmannian of complex "oriented" (fixing a trivialization of the determinant bundle) Grassmannian. Is there also a Schubert decomposition for this version of Grassmannian? Do you know any reference which does this explicitly?

@jdc What is this other notion of cell decomposition? Can you give some references?

Thanks to all. After a second look it does look like that the author neglected to mention the 0-cell, as what he claims later using this fact hold when there is a 0-cell. However, I still don't see why there are no 2-cells for general $n$.

@Ian Agol Thanks so much! Please do come back! I would appreciate references/more details of the counterexamples.

But such a surface may not have the minimal $\chi_-$ among its homology class.

Thanks! But I actually don't understand what you said about (1) either. There is no $f$ in (1). Are you constructing one starting from a surface with minimal $\chi_-$ in the class $[\theta]$ somehow? If "tubbing" means internal connected sum, it would increase $\chi_-$ unless one of the components is a sphere.

Yes, I should have said "local extrema". I don't know what "the conclusion in the homotopy class of $f$" means, but I need that the fiber of $f$ with minimal $\chi_-$ (among all fibers of $f$) has the minimal $\chi_-$ among embedded surfaces representing the same homology class.

@Ryan Budney: Thanks! For (2): Think of the circle as $\mathbb{R}/\mathbb{Z}$, then given $f: Y\to \mathbb{R}/\mathbb{Z}$, $df$ is a closed 1-form.