Algebraic & Geometric Topology

Homological stability properties of spaces of rational $J$–holomorphic curves in $\mathbb{P}^2$

Jeremy Miller

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Abstract

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.

Article information

Source
Algebr. Geom. Topol., Volume 13, Number 1 (2013), 453-478.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513715503

Digital Object Identifier
doi:10.2140/agt.2013.13.453

Mathematical Reviews number (MathSciNet)
MR3031648

Zentralblatt MATH identifier
1276.53083

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. Homological stability properties of spaces of rational $J$–holomorphic curves in $\mathbb{P}^2$. Algebr. Geom. Topol. 13 (2013), no. 1, 453--478. doi:10.2140/agt.2013.13.453. https://projecteuclid.org/euclid.agt/1513715503


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References

  • M Abreu, Topology of symplectomorphism groups of $S^2\times S^2$, Invent. Math. 131 (1998) 1–23
  • C P Boyer, B M Mann, Monopoles, nonlinear $\sigma$ models, and two-fold loop spaces, Comm. Math. Phys. 115 (1988) 571–594
  • F R Cohen, R L Cohen, B M Mann, R J Milgram, The topology of rational functions and divisors of surfaces, Acta Math. 166 (1991) 163–221
  • F R Cohen, R L Cohen, B M Mann, R J Milgram, The homotopy type of rational functions, Math. Z. 213 (1993) 37–47
  • M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307–347
  • H Hofer, V Lizan, J-C Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal. 7 (1997) 149–159
  • S Ivashkovich, V Shevchishin, Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls, Invent. Math. 136 (1999) 571–602
  • J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer, Berlin (1972)
  • D McDuff, The local behaviour of holomorphic curves in almost complex 4–manifolds, J. Differential Geom. 34 (1991) 143–164
  • D McDuff, D Salamon, $J$–holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society (1994)
  • G Segal, The topology of spaces of rational functions, Acta Math. 143 (1979) 39–72
  • J-C Sikorav, The gluing construction for normally generic $J$–holomorphic curves, from: “Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001)”, Fields Inst. Commun. 35, Amer. Math. Soc., Providence, RI (2003) 175–199