# memo_has_moved_text();Combinatorial Floer homology

Vin de Silva, Joel W. Robbin and Dietmar A. Salamon

Publication: Memoirs of the American Mathematical Society
Publication Year: 2014; Volume 230, Number 1080
ISBNs: 978-0-8218-9886-4 (print); 978-1-4704-1670-6 (online)
DOI: http://dx.doi.org/10.1090/memo/1080
Published electronically: December 10, 2013

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Chapters

• Chapter 1. Introduction
• Chapter 2. Chains and Traces
• Chapter 3. The Maslov Index
• Chapter 4. The Simply Connected Case
• Chapter 5. The Non Simply Connected Case
• Chapter 6. Lunes and Traces
• Chapter 7. Arcs
• Chapter 8. Combinatorial Lunes
• Chapter 9. Combinatorial Floer Homology
• Chapter 10. Hearts
• Chapter 11. Invariance under Isotopy
• Chapter 12. Lunes and Holomorphic Strips
• Chapter 13. Further Developments
• Appendix A. The Space of Paths
• Appendix B. Diffeomorphisms of the Half Disc
• Appendix C. Homological Algebra
• Appendix D. Asymptotic behavior of holomorphic strips

### Abstract

We define combinatorial Floer homology of a transverse pair of noncontractible nonisotopic embedded loops in an oriented $2$-manifold without boundary, prove that it is invariant under isotopy, and prove that it is isomorphic to the original Lagrangian Floer homology. Our proof uses a formula for the Viterbo-Maslov index for a smooth lune in a $2$-manifold.