## 4-manifolds (sympletic or not)

*Ana Cannas da Silva (Instituto Superior Tecnico, Lisboa)*

### Course description

We will review the state of the art and open questions on (smooth) 4-manifolds. We will explain the existence on 4-manifolds of structures related to symplectic forms.

### Tentative plan:

#### 1^{st} lecture - 4-Manifolds

Whereas (closed simply connected) topological 4-manifolds are completely classified, the panoramam for smooth 4-manifolds is quite wild: the existence of a smooth structure imposes strong topological constraints, yet for the same topology there can be infinite different smooth structures. We will discuss constructions of (smooth) 4-manifolds and invariants to distinguish them.

#### 2nd lecture - Symplectic Geography

Symplectic geography is concerned with existence and uniqueness of symplectic forms on a given manifold. These questions are particularly relevant to 4-dimensional topology and to mathematical physics, where symplectic manifolds occur as building blocks or as key examples. We will describe some constructions of symplectic 4-manifolds and invariants to distinguish them.

#### 3rd lecture - Folded Symplectic Forms

Any orientable 4-manifold admits a folded symplectic form, that is, a closed 2-form which is symplectic except on a separating hypersurface where the form singularities are like the pullback of a symplectic form by a folding map. We will explain how, for orientable even-dimensional manifolds, the existence of a stable almost complex structure is necessary and sufficient to warrant the existence of a folded symplectic form.