Homological stability properties of spaces of rational $J$–holomorphic curves in $\mathbb{P}^2$

Abstract

In a well known work, Graeme Segal proved 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. In this paper, we address if a similar result holds when other (not necessarily integrable) almost complex structures are put on projective space. We take almost complex structures that are compatible with the underlying symplectic structure. We obtain the following result: the inclusion of the space of based degree–$k$ $J$–holomorphic maps from ${\mathbb{P}}^1$ to ${\mathbb{P}}^2$ into the double loop space of ${\mathbb{P}}^2$ is a homology surjection for dimensions $j\le 3k-3$. The proof involves constructing a gluing map analytically in a way similar to McDuff and Salamon, and Sikorav, and then comparing it to a combinatorial gluing map studied by Cohen, Cohen, Mann, and Milgram.