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

Jeremy Miller

Geometry & Topology 19 (2015) 1829–1894

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.

almost-complex structure, little disks operad, gluing
Mathematical Subject Classification 2010
Primary: 53D05
Secondary: 55P48
Received: 21 November 2012
Revised: 6 October 2014
Accepted: 4 November 2014
Published: 29 July 2015
Proposed: Yasha Eliashberg
Seconded: Benson Farb, Leonid Polterovich
Jeremy Miller
Department of Mathematics
Stanford University
Building 380, Room 383A
Stanford, CA 94305