The topology of the space ofJ–holomorphic maps to ℂP2

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.