Smooth versus symplectic group actions on manifolds

Krzysztof Pawałowski, Adam Mickiewicz University

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Passcode: 998749

Until 2012, it was an open question whether on a closed smooth manifold admitting a symplectic form, there exists a smooth non-symplectic action of a compact Lie group, no matter what is the choice of the form.

In 2012, Kaluba and Politarczyk proved that any compact Lie group G can act smoothly on a complex projective space so that the fixed point set is not a symplectic manifold and thus, the action is not symplectic, regardless of the choice of symplectic form on the complex projective space.

For a finite group G not of prime power order, Oliver (1996) has answered the question which smooth manifolds M are diffeomorphic to the fixed point sets of smooth actions of G on disks and Euclidean spaces. Earlier, for a finite p-group G with p prime, this question has been answered for smooth actions of G on disks (Jones, 1971) and Euclidean spaces (kpa, 1982).

Our joint work with Hajduk (2020–2021) focuses on symplectic actions. Recently, we proved that for a finite group G, a compact exact symplectic manifold M with boundary of contact type is symplectomorphic to the fixed point set of a symplectic action of G on a symplectic disk, if and only if M is diffeomorphic to the fixed point set of a smooth action of G on a disk.

During my talk, I will recall the results of Jones (1971), kpa (1982), and Oliver (1996). Moreover, I will describe basic ideas which allow to perform the construction of smooth non-symplectic actions of Kaluba–Politarczyk (2012), and to obtain symplectic actions on disks of Hajduk–kpa (2021).