Homological stability properties of spaces of rational J–holomorphic curves in ℙ2

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 $ℙ^{1}$ to $ℙ^{2}$ into the double loop space of $ℙ^{2}$ is a homology surjection for dimensions $j \leq 3 k - 3$ . 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.

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Keywords

almost complex structure, little disks operad, gluing

Mathematical Subject Classification 2000

Primary: 53D05

Secondary: 55P48

References

Publication

Received: 10 February 2012

Revised: 15 October 2012

Accepted: 16 October 2012

Published: 6 March 2013

Authors

Jeremy Miller
Department of Mathematics CUNY Graduate Center 365 Fifth Avenue New York, NY 10016 USA
http://https://sites.google.com/site/jeremykennethmiller/

Volume 13, issue 1 (2013)

Jeremy Miller

Abstract