Flexible Surfaces in $\mathbb{C}P^2$ and $S^2\times S^2$

TL;DR

This paper introduces the concept of flexible surfaces in 4-manifolds, constructing examples in complex projective planes and product spaces, and explores obstructions and special cases related to spin structures.

Contribution

It defines flexibility for surfaces in 4-manifolds and constructs flexible and spin-flexible surfaces in specific manifolds within various homology classes.

Findings

Flexible surfaces exist in $\, ext{CP}^2$ and $S^2 imes S^2$.

Obstructions due to spin structures are identified for characteristic classes.

Spin-flexible representatives are constructed despite obstructions.

Abstract

A surface $\Sigma$ in a 4-manifold $M$ is called flexible if any mapping class of the surface arises as the restriction of a diffeomorphism $(M,\Sigma) \to (M,\Sigma)$. We construct flexible surfaces in $\mathbb{C}P^2$ and $S^2 \times S^2$ within any prescribed non-characteristic homology class. Within characteristic homology classes there is a spin structure obstructing flexibility and we construct so-called spin-flexible representatives.