Almost Complex Structures on $S^2\times S^2$

TL;DR

This paper studies the topology and stratification of the space of compatible almost complex structures on $S^2\times S^2$, revealing intricate links and their relation to symplectomorphism groups.

Contribution

It characterizes the stratification of the space of compatible almost complex structures on $S^2\times S^2$ as finite codimension Fr\'echet manifolds with complex links, extending to other ruled surfaces.

Findings

Strata are finite codimension Fr\'echet manifolds.

Normal links are finite dimensional stratified spaces.

Topology of links is complex and case-dependent.

Abstract

In this note we investigate the structure of the space $\Jj$ of smooth almost complex structures on $S^2\times S^2$ that are compatible with some symplectic form. This space has a natural stratification that changes as the cohomology class of the form changes and whose properties are very closely connected to the topology of the group of symplectomorphisms of $S^2\times S^2$. By globalizing standard gluing constructions in the theory of stable maps, we show that the strata of $\Jj$ are Fr\'echet manifolds of finite codimension, and that the normal link of each stratum is a finite dimensional stratified space. The topology of these links turns out to be surprisingly intricate, and we work out certain cases. Our arguments apply also to other ruled surfaces, though they give complete information only for bundles over $S^2$ and $T^2$.