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Combinatorial Floer Homology
Vin de Silva, Pomona College, Claremont, CA, Joel W. Robbin, University of Wisconsin, Madison, WI, and Dietmar A. Salamon, ETH Zurich, Switzerland
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Memoirs of the American Mathematical Society
2014; 114 pp; softcover
Volume: 230
ISBN-10: 0-8218-9886-8
ISBN-13: 978-0-8218-9886-4
List Price: US$75 Individual Members: US$45
Institutional Members: US\$60
Order Code: MEMO/230/1080

The authors 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. Their proof uses a formula for the Viterbo-Maslov index for a smooth lune in a $$2$$-manifold.

• Introduction
Part I. The Viterbo-Maslov Index
• Chains and traces
• The Maslov index
• The simply connected case
• The Non simply connected case
Part II. Combinatorial Lunes
• Lunes and traces
• Arcs
• Combinatorial lunes
Part III. Floer Homology
• Combinatorial Floer homology
• Hearts
• Invariance under isotopy
• Lunes and holomorphic strips
• Further developments
Appendices
• Appendix A. The space of paths
• Appendix B. Diffeomorphisms of the half disc
• Appendix C. Homological algebra
• Appendix D. Asymptotic behavior of holomorphic strips
• Bibliography
• Index