Homological stability properties of spaces of rational J-holomorphic curves in P^2

TL;DR

This paper investigates the homological properties of spaces of rational J-holomorphic curves in P^2, extending classical results to non-integrable almost complex structures compatible with symplectic forms.

Contribution

It demonstrates that the inclusion of the space of degree k J-holomorphic maps into the double loop space of P^2 induces a homology surjection in a specific dimension range, generalizing Segal's theorem.

Findings

Homology surjection for dimensions j<3k-2

Construction of an analytical gluing map

Comparison with combinatorial gluing map

Abstract

In a well known work [Se], Graeme Segal proved that the space of holomorphic maps from a Riemann surface to a complex projective space is homology equivalent to the corresponding continuous mapping space 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 P^1 to P^2 into the double loop space of P^2 is a homology surjection for dimensions j<3k-2. The proof involves constructing a gluing map analytically in a way similar to McDuff and Salamon in [MS] and Sikorav in [S] and then comparing it to a combinatorial gluing map studied by Cohen, Cohen, Mann, and Milgram in [CCMM].