The topology of the space of $J$–holomorphic maps to $\mathbb{C}\mathrm{P}^2$

Abstract

The purpose of this paper is to generalize a theorem of Segal 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 $ℂP^{\mathfrak{2}}$, 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 with analytic gluing maps for $J$–holomorphic curves . This is an extension of the author’s work regarding genus-zero case.