Note on coisotropic Floer homology and leafwise fixed points

Fabian Ziltener; Fabian Ziltener

doi:10.3934/era.2021001

Electronic Research Archive

2021, Volume 29, Issue 4: 2553-2560. doi: 10.3934/era.2021001

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Note on coisotropic Floer homology and leafwise fixed points

Fabian Ziltener

Utrecht University, Mathematics Institute, Budapestlaan 6, 3584 CD Utrecht, The Netherlands

Received: 01 November 2019 Revised: 01 September 2020 Published: 11 January 2021
Primary: 53D40

For an adiscal or monotone regular coisotropic submanifold $ N $ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $ N $. Given a Hamiltonian isotopy $ \varphi = ( \varphi^t) $ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $ (N, \varphi) $-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.
Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.
The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.
- Floer homology,
- coisotropic submanifold,
- leafwise fixed point,
- leafwise intersection,
- translated point
Citation: Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points[J]. Electronic Research Archive, 2021, 29(4): 2553-2560. doi: 10.3934/era.2021001

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Abstract

For an adiscal or monotone regular coisotropic submanifold $ N $ of a symplectic manifold I define its Floer homology to be the Floer homology of a certain Lagrangian embedding of $ N $. Given a Hamiltonian isotopy $ \varphi = ( \varphi^t) $ and a suitable almost complex structure, the corresponding Floer chain complex is generated by the $ (N, \varphi) $-contractible leafwise fixed points. I also outline the construction of a local Floer homology for an arbitrary closed coisotropic submanifold.

Results by Floer and Albers about Lagrangian Floer homology imply lower bounds on the number of leafwise fixed points. This reproduces earlier results of mine.

The first construction also gives rise to a Floer homology for a Boothby-Wang fibration, by applying it to the circle bundle inside the associated complex line bundle. This can be used to show that translated points exist.

References

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