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Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.12 no.3 Temuco  2010

http://dx.doi.org/10.4067/S0719-06462010000300004 

CUBO A Mathematical Journal Vol.12, N° 03, (49–69). October 2010

 

On The Group of Strong Symplectic Homeomorphisms

 

AUGUSTIN BANYAGA

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 email: banyaga@math.psu.edu


ABSTRACT

We generalize the “hamiltonian topology” on hamiltonian isotopies to an intrinsic “symplectic topology” on the space of symplectic isotopies. We use it to define the group SSympeo (M,ω) of strong symplectic homeomorphisms, which generalizes the group Hameo(M,ω) of hamiltonian homeomorphisms introduced by Oh and Müller. The group SSympeo(M,ω) is arcwise connected, is contained in the identity component of Sympeo(M,ω); it contains Hameo(M,ω) as a normal subgroup and coincides with it when M is simply connected. Finally its commutator subgroup [SSympeo(M,ω), SSympeo(M,ω)] is contained in Hameo(M,ω).

Key words and phrases: Hamiltonian homeomorphisms, hamiltonian topology, symplectic topology, stromg symplectic homeomorphisms, C0 symplectic topology.


RESUMEN

Generalizamos la “topología hamiltoniano” sobre isotopias hamiltonianas para una “topología simpléctica” intrinseca en el espacio de isotopias simplécticas. Nosotros usamos esto para definir el grupo SSympeo(M,ω) de homeomorfismos simplécticos fuertes, el qual generaliza el grupo Hameo(M,ω) de homeomorfismos hamiltonianos introducido por Oh y Müller. El grupo SSympeo(M,ω) es conexo por arcos, es contenido en la componente identidad de Sympeo(H,ω); este contiene Hameo(M,ω) como un subgrupo normal y coincide con este cuando M es simplemente conexa. Finalmente su subgrupo conmutador [SSympeo(M,ω), SSympeo(M,ω)] es contenido en Hameo(M,ω).

Math. Subj. Class.: MSC2000:53D05; 53D35.


Acknowledgement

I would like to thank Claudio Cuevas for soliciting this paper for Cubo.

I am also very grateful to the referee for an extensive list of good remarks, questions, and suggestions which drastically improved the final form of this paper. In particular I owe to him/her the idea to finish the Proof of Theorem 1.

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