# A Lagrangian Piunikhin-Salamon-Schwarz Morphism and Two Comparison Homomorphisms in Floer Homology

@article{Albers2005ALP, title={A Lagrangian Piunikhin-Salamon-Schwarz Morphism and Two Comparison Homomorphisms in Floer Homology}, author={Peter Albers}, journal={International Mathematics Research Notices}, year={2005}, volume={2008} }

In this article we address two issues. First, we explore to what extent the techniques of Piunikhin, Salamon and Schwarz in [PSS96] can be carried over to Lagrangian Floer homology. In [PSS96] an isomorphism between Hamiltonian Floer homology and the singular homology is established. In contrast, Lagrangian Floer homology is not isomorphic to the singular homology of the Lagrangian submanifold, in general. Depending on the minimal Maslov number, we construct for certain degrees two…

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## References

SHOWING 1-10 OF 27 REFERENCES

### Cohomology operations from S¹-cobordisms in Floer homology

- Mathematics
- 1995

In this work, Floer homology is considered as a relative Morse theory for the symplectic action functional on the loop space of a symplectic manifold (M, to). It is assumed that M is closed and the…

### NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY

- Mathematics
- 2006

Given the cotangent bundle T ∗ Q of a smooth manifold with its canonical symplectic structure, and a Hamiltonian function on T ∗ Q which is fiberwise asymptot- ically quadratic, its well-defined…

### J-Holomorphic Curves and Symplectic Topology

- Mathematics
- 2004

The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was…

### On the extrinsic topology of Lagrangian submanifolds

- Mathematics
- 2005

We investigate the extrinsic topology of Lagrangian submanifolds and of their submanifolds in closed symplectic manifolds using Floer homological methods. The first result asserts that the homology…

### Symplectic Floer-Donaldson theory and quantum cohomology

- Mathematics
- 1996

The goal of this paper is to give in outline a new proof of the fact that the Floer cohomology groups of the loop space of a semi-positive symplectic manifold (M; !) are naturally isomorphic to the…

### Floer homology and Novikov rings

- Mathematics
- 1995

We prove the Arnold conjecture for compact symplectic manifolds under the assumption that either the first Chern class of the tangent bundle vanishes over π2(M) or the minimal Chern number is at…

### Morse theory for periodic solutions of hamiltonian systems and the maslov index

- Mathematics
- 1992

In this paper we prove Morse type inequalities for the contractible 1-periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the…

### Addendum to “Floer Cohomology of Lagrangian Intersections and Pseudo-Holomorphic Discs, I”

- Mathematics
- 1995

This is an addendum to the author's earlier paper “Floer Cohomology of Lagrangian Intersection and Pseudo-Holomorphic Discs, I,” Comm. Pure Appl. Math. 46, 1993, pp. 949–993. The main result of this…

### Transversality in elliptic Morse theory for the symplectic action

- Mathematics
- 1995

Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder. These results play a central role in the denition of symplectic…

### Symplectic fixed points and holomorphic spheres

- Mathematics
- 1989

LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the…