Symplectic $(-2)$-spheres and the symplectomorphism group of small rational 4-manifolds II
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- by Jun Li, Tian-Jun Li and Weiwei Wu PDF
- Trans. Amer. Math. Soc. 375 (2022), 1357-1410 Request permission
Abstract:
We study the symplectic mapping class groups of $(\mathbb {C} P^2 \# 5{\overline {\mathbb {C} P^2}},\omega )$. Our main innovation is to avoid the detailed analysis of the topology of generic almost complex structures $\mathcal {J}_0$ as in most of earlier literature. Instead, we use a combination of the technique of ball-swapping (defined by Wu [Math. Ann. 359 (2014), pp. 153–168]) and the study of a semi-toric model to understand a “connecting map”, whose cokernel is the symplectic mapping class group.
Using this approach, we completely determine the Torelli symplectic mapping class group (Torelli SMCG) for all symplectic forms $\omega$. Let $N_{\omega }$ be the number of $(-2)$-symplectic spherical homology classes. Torelli SMCG is trivial if $N_{\omega }>8$; it is $\pi _0(\text {Diff}^+(S^2,5))$ if $N_{\omega }=0$ (by Seidel [Lecture notes in Math., Springer, Berlin, 2008] and Evans [J. Symplectic Geom. 9 (2011), pp. 45–82]); and it is $\pi _0(\text {Diff}^+(S^2,4))$ in the remaining case. Further, we completely determine the rank of $\pi _1(Symp(\mathbb {C} P^2 \# 5{\overline {\mathbb {C} P^2}},\omega ))$ for any given symplectic form. Our results can be uniformly presented in terms of Dynkin diagrams of type $\mathbb {A}$ and type $\mathbb {D}$ Lie algebras. We also provide a solution to the smooth isotopy problem of rational 4-manifolds (open problem 16 in McDuff-Salamon’s book 3rd version [Oxford graduate texts in mathematics, Oxford University Press, Oxford, 2017]).
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Additional Information
- Jun Li
- Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
- MR Author ID: 1113604
- ORCID: 0000-0003-1655-0472
- Email: lijungeo@umich.edu
- Tian-Jun Li
- Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
- MR Author ID: 358138
- Email: tjli@math.umn.edu
- Weiwei Wu
- Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
- MR Author ID: 962029
- Email: weiwei.wu@uga.edu
- Received by editor(s): February 10, 2020
- Received by editor(s) in revised form: June 12, 2021, and July 8, 2021
- Published electronically: November 5, 2021
- Additional Notes: The first author was supported by NSF Grants and an AMS-Simons travel grant. The second author was supported by NSF Grants. The third author was partially supported by Simons Collaboration Grant 524427.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 1357-1410
- MSC (2020): Primary 57R17, 53D35; Secondary 14D22, 57S05
- DOI: https://doi.org/10.1090/tran/8517
- MathSciNet review: 4369250