Geometry & Topology

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

Jeremy Miller

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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 P2, 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.

Article information

Source
Geom. Topol., Volume 19, Number 4 (2015), 1829-1894.

Dates
Received: 21 November 2012
Revised: 6 October 2014
Accepted: 4 November 2014
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1510858797

Digital Object Identifier
doi:10.2140/gt.2015.19.1829

Mathematical Reviews number (MathSciNet)
MR3375520

Zentralblatt MATH identifier
1332.32032

Subjects
Primary: 53D05: Symplectic manifolds, general
Secondary: 55P48: Loop space machines, operads [See also 18D50]

Keywords
almost-complex structure little disks operad gluing

Citation

Miller, Jeremy. The topology of the space of $J$–holomorphic maps to $\mathbb{C}\mathrm{P}^2$. Geom. Topol. 19 (2015), no. 4, 1829--1894. doi:10.2140/gt.2015.19.1829. https://projecteuclid.org/euclid.gt/1510858797


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