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2013 Homological stability properties of spaces of rational $J$–holomorphic curves in $\mathbb{P}^2$
Jeremy Miller
Algebr. Geom. Topol. 13(1): 453-478 (2013). DOI: 10.2140/agt.2013.13.453


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 1 to 2 into the double loop space of 2 is a homology surjection for dimensions j3k3. 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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Jeremy Miller. "Homological stability properties of spaces of rational $J$–holomorphic curves in $\mathbb{P}^2$." Algebr. Geom. Topol. 13 (1) 453 - 478, 2013.


Received: 10 February 2012; Revised: 15 October 2012; Accepted: 16 October 2012; Published: 2013
First available in Project Euclid: 19 December 2017

zbMATH: 1276.53083
MathSciNet: MR3031648
Digital Object Identifier: 10.2140/agt.2013.13.453

Primary: 53D05
Secondary: 55P48

Keywords: almost complex structure , gluing , little disks operad

Rights: Copyright © 2013 Mathematical Sciences Publishers


Vol.13 • No. 1 • 2013
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