July  2008, 2(3): 431-455. doi: 10.3934/jmd.2008.2.431

Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds


Fields Institute, Toronto, M5T 3J1, Canada

Received  October 2007 Revised  February 2008 Published  April 2008

We prove that the group of Hamiltonian automorphisms of a symplectic $4$-manifold $(M,\omega)$, Ham$(M,\omega)$, contains only finitely many conjugacy classes of maximal compact tori with respect to the action of the full symplectomorphism group Symp$(M,\omega)$. We also consider the set of conjugacy classes of\/ $2$-tori in Ham$(M,\omega)$ with respect to Hamiltonian conjugation and show that its finiteness is equivalent to the finiteness of the symplectic mapping class group $\pi_{0}$(Symp$(M,\omega)$). Finally, we extend to rational and ruled manifolds a result of Kedra which asserts that if $(M,\omega)$ is a simply connected symplectic $4$-manifold with $b_{2}\geq 3$, and if $(\widetilde{M},\widetilde{\omega}_{\delta})$ denotes a symplectic blow-up of $(M,\omega)$ of small enough capacity $\delta$, then the rational cohomology algebra of the Hamiltonian group Ham($\widetilde{M},\widetilde{\omega}_{\delta})$ is not finitely generated. Our results are based on the fact that in a symplectic $4$-manifold endowed with any tamed almost structure $J$, exceptional classes of minimal symplectic area are $J$-indecomposable.
Citation: Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431

Grzegorz Graff, Jerzy Jezierski. Minimization of the number of periodic points for smooth self-maps of closed simply-connected 4-manifolds. Conference Publications, 2011, 2011 (Special) : 523-532. doi: 10.3934/proc.2011.2011.523


Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537


Benjamin Weiss. Entropy and actions of sofic groups. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3375-3383. doi: 10.3934/dcdsb.2015.20.3375


Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443


Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63.


Michel Coornaert, Fabrice Krieger. Mean topological dimension for actions of discrete amenable groups. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 779-793. doi: 10.3934/dcds.2005.13.779


Nancy Guelman, Isabelle Liousse. Actions of Baumslag-Solitar groups on surfaces. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1945-1964. doi: 10.3934/dcds.2013.33.1945


Richard Miles, Michael Björklund. Entropy range problems and actions of locally normal groups. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 981-989. doi: 10.3934/dcds.2009.25.981


Andrei Török. Rigidity of partially hyperbolic actions of property (T) groups. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 193-208. doi: 10.3934/dcds.2003.9.193


Jan J. Dijkstra and Jan van Mill. Homeomorphism groups of manifolds and Erdos space. Electronic Research Announcements, 2004, 10: 29-38.


Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031


Pablo Angulo. Linking curves, sutured manifolds and the Ambrose conjecture for generic 3-manifolds. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 1-41. doi: 10.3934/dcds.2018001


Anatole Katok, Federico Rodriguez Hertz. Arithmeticity and topology of smooth actions of higher rank abelian groups. Journal of Modern Dynamics, 2016, 10: 135-172. doi: 10.3934/jmd.2016.10.135


Brandon Seward. Krieger's finite generator theorem for actions of countable groups Ⅱ. Journal of Modern Dynamics, 2019, 15: 1-39. doi: 10.3934/jmd.2019012


Juan Alonso, Nancy Guelman, Juliana Xavier. Actions of solvable Baumslag-Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1817-1827. doi: 10.3934/dcds.2015.35.1817


Alexander Gorodnik, Theron Hitchman, Ralf Spatzier. Regularity of conjugacies of algebraic actions of Zariski-dense groups. Journal of Modern Dynamics, 2008, 2 (3) : 509-540. doi: 10.3934/jmd.2008.2.509


Danijela Damjanovic and Anatole Katok. Local rigidity of actions of higher rank abelian groups and KAM method. Electronic Research Announcements, 2004, 10: 142-154.


Cristóbal Camarero, Carmen Martínez, Ramón Beivide. Identifying codes of degree 4 Cayley graphs over Abelian groups. Advances in Mathematics of Communications, 2015, 9 (2) : 129-148. doi: 10.3934/amc.2015.9.129


Dwayne Chambers, Erica Flapan, John D. O'Brien. Topological symmetry groups of $K_{4r+3}$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1401-1411. doi: 10.3934/dcdss.2011.4.1401


Fei Liu, Jaume Llibre, Xiang Zhang. Heteroclinic orbits for a class of Hamiltonian systems on Riemannian manifolds. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1097-1111. doi: 10.3934/dcds.2011.29.1097

2020 Impact Factor: 0.848


  • PDF downloads (146)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]