The topology of the space of J–holomorphic maps to ℂP2

Miller, Jeremy

doi:10.2140/gt.2015.19.1829

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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

DOI: 10.2140/gt.2015.19.1829

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 $ℂ P^{2}$ , 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.

Keywords

almost-complex structure, little disks operad, gluing

Mathematical Subject Classification 2010

Primary: 53D05

Secondary: 55P48

References

Publication

Received: 21 November 2012

Revised: 6 October 2014

Accepted: 4 November 2014

Published: 29 July 2015

Proposed: Yasha Eliashberg

Seconded: Benson Farb, Leonid Polterovich

Authors

Jeremy Miller
Department of Mathematics Stanford University Building 380, Room 383A Stanford, CA 94305 USA
http://math.stanford.edu/~jkmiller/

Volume 19, issue 4 (2015)