Celebratio Mathematica — McDuff

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Works connected to Miguel Tribolet Abreu

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M. Abreu and D. McDuff: “Topology of symplectomorphism groups of rational ruled surfaces,” J. Am. Math. Soc. 13 : 4 (2000), pp. 971–1009. MR 1775741 Zbl 0965.57031 article

Let \( M \) be either \( S^2\times S^2 \) or the one point blow-up \( \mathbb{C}P^2\# \overline{\mathbb{C}P}^2 \) of \( \mathbb{C}P^2 \). In both cases \( M \) carries a family of symplectic forms \( \omega_{\lambda} \), where \( \lambda > -1 \) determines the cohomology class \( [\omega_\lambda] \). This paper calculates the rational (co)homology of the group \( G_\lambda \) of symplectomorphisms of \( (M,\omega_\lambda) \) as well as the rational homotopy type of its classifying space \( BG_\lambda \). It turns out that each group \( G_\lambda \) contains a finite collection \( K_k \), \( k = 0 \),\( \dots,\ell = \ell(\lambda) \), of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups “asymptotically commute”, i.e. all the higher Whitehead products that they generate vanish as \( \lambda\to \infty \). However, for each fixed \( \lambda \) there is essentially one nonvanishing product that gives rise to a “jumping generator” \( w_\lambda \) in \( H^*(G_\lambda) \) and to a single relation in the rational cohomology ring \( H^*(BG_\lambda) \). An analog of this generator \( w_\lambda \) was also seen by Kronheimer in his study of families of symplectic forms on 4-manifolds using Seiberg–Witten theory. Our methods involve a close study of the space of \( \omega_\lambda \)-compatible almost complex structures on \( M \).

M. Abreu, M. S. Borman, and D. McDuff: “Displacing Lagrangian toric fibers by extended probes,” Algebr. Geom. Topol. 14 : 2 (2014), pp. 687–752. MR 3159967 Zbl 1288.53076 article

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff’s original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space \( \mathbb{P}(1,3,5) \) after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

Matthew Strom Borman	Related
Miguel Tribolet Abreu	Related