Abstract:

symplectic manifold is a pair $\left(X^{2n},\omega\right)$, where $X^{2n}$ is a smooth manifold and $\omega$ is a differential 2-form such that $\operatorname{d}\omega=0$ and $\omega^n>0$, known as the symplectic form. This simple definition gives rise to a broad area in geometry and topology with many connections to other disciplines such as classical mechanics, low-dimensional topology or algebraic and complex geometry. Among the many objects that one can study in this setting, we find the Lagrangian submanifolds. These are those submanifolds $L$ of half the ambient dimension on which the symplectic form vanishes identically on each tangent space of $L$. The study of Lagrangian submanifolds is a central topic in symplectic topology that can tell us a great deal about the symplectic manifold $(X,\omega)$. There are many interesting questions one can ask about Lagrangian submanifolds. In this work, we will study one of these.

We study the minimal genus question for a symplectic rational 4-manifold $(X,\omega)$, which asks, for a given $A\in H_2(X;\Z_2)$, what are the possible topological types of non-orientable Lagrangian surfaces in the class $A$; and specially, what is the maximal Euler number, or, equivalently, the minimal genus. We start by ensuring that 2-homology classes can be represented by a non-orientable surface. Next, we are able to proof that, when having a symplectic structure in our manifold, these surfaces representing the homology classes can be taken to be non-orientable embedded Lagrangians. In this setup, the minimal genus question arises, and we study a partial answer to this question for rational 4-manifolds. We will see that if a homology class $A\in H_2(X;\Z_2)$ is realised by a non-orientable embedded Lagrangian surface $L$, then $\mathcal{P}(A)=\chi(L)\ \operatorname{mod}\ 4$, where $\mathcal{P}(A)$ is the Pontrjagin square of $A$. We will briefly discuss the problem for the zero class, and prove the main result of the essay for non-zero classes, which states the reciprocate for some symplectic structures in the case of rational 4-manifolds.