Abstract
We prove the stability of \(\textrm{Symp}(X,w)\cap \textrm{Diff}_0(X)\) for a one-point blow-up of irrational ruled surfaces and study their topological colimit. Non-trivial generators of \(\pi _0[\textrm{Symp}(X,w)\cap \textrm{Diff}_0(X)]\) that differ from Lagrangian Dehn twists are detected.
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Notes
The components of \([\mu , 1, c_1, \ldots , c_n)] \) are the areas of the homology class \(B, F, E_1, \ldots , E_n\), where \(\mu >0, c_1+c_2<1, 0<c_i<1,\) \( c_1\ge c_2\ge \cdots \ge c_n, 2 \mu > c_1^2+ \ldots c_n^2 \).
McDuff’s original results were written in terms of homotopy fibrations where the larger space of taming almost complex structures was used. Since this space is homotopy equivalent to the space of compatible structures we use (for reasons explained in Sect. 3.4) we will use the compatible structure spaces instead.
The inflation along \(B+xF\) could be achieved using curves \(B+xF-E\) as as in Lemma 4.4.
A convenient way of avoiding the \(\epsilon , \delta \) in each step of the J-tame inflation process is to use the formal inflation as defined in [30].
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Acknowledgements
Jun Li is supported by an AMS-Simons travel grant. We are grateful to Richard Hind, Tian-Jun Li, Dusa McDuff, Weiwei Wu, Weiyi Zhang for helpful conversations. We thank an anonymous referee for his/her very careful reading and many helpful suggestions throughout the paper.
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Buse, O., Li, J. Symplectic isotopy on non-minimal ruled surfaces. Math. Z. 304, 44 (2023). https://doi.org/10.1007/s00209-023-03298-3
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DOI: https://doi.org/10.1007/s00209-023-03298-3