# Non-orientable Lagrangian surfaces in rational 4-manifolds

## TL;DR

This paper characterizes when nonorientable Lagrangian surfaces in rational 4-manifolds exist for given homology classes, linking their existence to the Pontrjagin square mod 4.

## Contribution

It provides a necessary and sufficient condition for representing homology classes by nonorientable Lagrangian surfaces in rational 4-manifolds.

## Key findings

- Characterization of classes represented by nonorientable Lagrangian surfaces
- Relation between Pontrjagin square and Lagrangian surface existence
- Condition involving mod 4 Pontrjagin square

## Abstract

We show that for any nonzero class $A$ in $H_2(X; \mathbb{Z}_2)$ in a rational 4-manifold $X$, $A$ is represented by a nonorientable embedded Lagrangian surface L (for some symplectic structure) if and only if $P(A)\equiv (L) (mod\ 4)$; where $P(A)$ denotes the mod 4 valued Pontrjagin square of $A$.