Topology of spaces of equivariant symplectic embeddings

Pelayo, Alvaro

doi:10.1090/S0002-9939-06-08310-9

Topology of spaces of equivariant symplectic embeddings
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by Alvaro Pelayo

Proc. Amer. Math. Soc. 135 (2007), 277-288

DOI: https://doi.org/10.1090/S0002-9939-06-08310-9

Published electronically: July 28, 2006

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Abstract:

We compute the homotopy type of the space of $\mathbb {T}^n$-equivariant symplectic embeddings from the standard $2n$-dimensional ball of some fixed radius into a $2n$-dimensional symplectic–toric manifold $(M, \sigma )$, and use this computation to define a $\mathbb {Z}_{\ge 0}$-valued step function on $\mathbb {R}_{\ge 0}$ which is an invariant of the symplectic–toric type of $(M, \sigma )$. We conclude with a discussion of the partially equivariant case of this result.

References

M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15. MR 642416, DOI 10.1112/blms/14.1.1
Sílvia Anjos, Homotopy type of symplectomorphism groups of $S^2\times S^2$, Geom. Topol. 6 (2002), 195–218. MR 1914568, DOI 10.2140/gt.2002.6.195
Paul Biran, Connectedness of spaces of symplectic embeddings, Internat. Math. Res. Notices 10 (1996), 487–491. MR 1399413, DOI 10.1155/S1073792896000323
Paul Biran, From symplectic packing to algebraic geometry and back, European Congress of Mathematics, Vol. II (Barcelona, 2000) Progr. Math., vol. 202, Birkhäuser, Basel, 2001, pp. 507–524. MR 1909952
P. Biran, Geometry of symplectic intersections, Proc. ICM Beijing III (2004) 1-3.
Ana Cannas da Silva, Lectures on symplectic geometry, Lecture Notes in Mathematics, vol. 1764, Springer-Verlag, Berlin, 2001. MR 1853077, DOI 10.1007/978-3-540-45330-7
Thomas Delzant, Hamiltoniens périodiques et images convexes de l’application moment, Bull. Soc. Math. France 116 (1988), no. 3, 315–339 (French, with English summary). MR 984900
V. Guillemin and S. Sternberg, Convexity properties of the moment mapping, Invent. Math. 67 (1982), no. 3, 491–513. MR 664117, DOI 10.1007/BF01398933
Victor Guillemin and Shlomo Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1984. MR 770935
Victor Guillemin, Moment maps and combinatorial invariants of Hamiltonian $T^n$-spaces, Progress in Mathematics, vol. 122, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1301331, DOI 10.1007/978-1-4612-0269-1
Victor Guillemin, Viktor Ginzburg, and Yael Karshon, Moment maps, cobordisms, and Hamiltonian group actions, Mathematical Surveys and Monographs, vol. 98, American Mathematical Society, Providence, RI, 2002. Appendix J by Maxim Braverman. MR 1929136, DOI 10.1090/surv/098
M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307–347. MR 809718, DOI 10.1007/BF01388806
M. Hirsch, Differential topology. Springer-Verlag 33 (1980).
François Lalonde and Martin Pinsonnault, The topology of the space of symplectic balls in rational 4-manifolds, Duke Math. J. 122 (2004), no. 2, 347–397. MR 2053755, DOI 10.1215/S0012-7094-04-12223-7
Yael Karshon and Susan Tolman, Centered complexity one Hamiltonian torus actions, Trans. Amer. Math. Soc. 353 (2001), no. 12, 4831–4861. MR 1852084, DOI 10.1090/S0002-9947-01-02799-4
Y. Karshon and S. Tolman, The Gromov width of complex Grasmannians, Alg. and Geom. Topol. 5 paper 38, 911–922.
Dusa McDuff, From symplectic deformation to isotopy, Topics in symplectic $4$-manifolds (Irvine, CA, 1996) First Int. Press Lect. Ser., I, Int. Press, Cambridge, MA, 1998, pp. 85–99. MR 1635697
Dusa McDuff, The structure of rational and ruled symplectic $4$-manifolds, J. Amer. Math. Soc. 3 (1990), no. 3, 679–712. MR 1049697, DOI 10.1090/S0894-0347-1990-1049697-8
A. Pelayo, Toric symplectic ball packing, To appear in Topol. Appl..