The topology of the space of J-holomorphic maps to CP^2

TL;DR

This paper extends Segal's theorem to show that for any compatible almost complex structure on CP^2, the space of J-holomorphic maps approximates the space of continuous maps in homology, with the approximation improving as degree increases.

Contribution

It generalizes Segal's homology equivalence result from holomorphic to J-holomorphic maps on CP^2 for arbitrary compatible almost complex structures.

Findings

Homology surjection from J-holomorphic to continuous maps for CP^2.

Extension of previous genus zero results to broader almost complex structures.

Comparison of topological chiral homology with analytic gluing maps for J-holomorphic curves.

Abstract

The purpose of this paper is to generalize a theorem of Segal from [Seg79] proving that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding space of continuous maps through a range of dimensions increasing with degree. We will address if a similar result holds when other almost complex structures are put on a projective space. For any compatible almost complex structure J on CP^2, we prove that the inclusion map from the space of J-holomorphic maps to the space of continuous maps induces a homology surjection through a range of dimensions tending to infinity with degree. The proof involves comparing the scanning map of topological chiral homology [Lur09], [And10], [Mil12]) with analytic gluing maps for J-holomorphic curves ([MS94], [Sik03]). This is an extension of the work done in [Mil11] regarding genus zero case.