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April 2019 The Homotopy Lie Algebra of Symplectomorphism Groups of 3-Fold Blowups of (S2×S2,σstdσstd)
Sílvia Anjos, Sinan Eden
Michigan Math. J. 68(1): 71-126 (April 2019). DOI: 10.1307/mmj/1547089467


We consider the 3-point blowup of the manifold (S2×S2,σσ), where σ is the standard symplectic form that gives area 1 to the sphere S2, and study its group of symplectomorphisms Symp(S2×S2#3CP¯2,ω). So far, the monotone case was studied by Evans [6], who proved that this group is contractible. Moreover, Li, Li, and Wu [13] showed that the group Symph(S2×S2#3CP¯2,ω) of symplectomorphisms that act trivially on homology is always connected, and recently, in [14], they also computed its fundamental group. We describe, in full detail, the rational homotopy Lie algebra of this group.

We show that some particular circle actions contained in the symplectomorphism group generate its full topology. More precisely, they give the generators of the homotopy graded Lie algebra of Symp(S2×S2#3CP¯2,ω). Our study depends on Karshon’s classification of Hamiltonian circle actions and the inflation technique introduced by Lalonde and McDuff. As an application, we deduce the rank of the homotopy groups of Symp(CP2#5CP¯2,ω˜) in the case of small blowups.


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Sílvia Anjos. Sinan Eden. "The Homotopy Lie Algebra of Symplectomorphism Groups of 3-Fold Blowups of (S2×S2,σstdσstd)." Michigan Math. J. 68 (1) 71 - 126, April 2019.


Received: 15 February 2017; Revised: 1 February 2018; Published: April 2019
First available in Project Euclid: 10 January 2019

zbMATH: 07155459
MathSciNet: MR3934605
Digital Object Identifier: 10.1307/mmj/1547089467

Primary: 53D35
Secondary: 57R17 , 57S05 , 57T20

Rights: Copyright © 2019 The University of Michigan


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Vol.68 • No. 1 • April 2019
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