1. INTRODUCTION 3
isotopy invariance of combinatorial Floer homology by examining generic deforma-
tions of loops that change the number of intersection points. This is very much
in the spirit of Floer’s original proof of deformation invariance (under Hamiltonian
isotopy of the Lagrangian manifolds) of analytic Floer homology. The main theo-
rem in Chapter 12 asserts, in the general setting, that smooth lunes (up to isotopy)
are in one-to-one correspondence with index one holomorphic strips (up to transla-
tion). The proof is self-contained and does not use any of the other results in this
memoir. It is based on an equation (the index formula (69) in Theorem 12.2) which
expresses the Viterbo–Maslov index of a holomorphic strip in terms of its critical
points and its angles at infinity. A linear version of this equation (the linear index
formula (76) in Lemma 12.3) also shows that every holomorphic strip is regular in
the sense that the linearized operator is surjective. It follows from these observa-
tions that the combinatorial and analytic definitions of Floer homology agree as
asserted in Theorem 9.3. In fact, our results show that the two chain complexes
agree.
There are many directions in which the theory developed in the present memoir
can be extended. Some of these are discussed in Chapter 13. For example, it has
been understood for some time that the Donaldson triangle product and the Fukaya
category have combinatorial analogues in dimension two, and that these analogues
are isomorphic to the original analytic theories. The combinatorial approach to
the Donaldson triangle product has been outlined in the PhD thesis of the first
author [6], and the combinatorial approach to the derived Fukaya category has
been used by Abouzaid [1] to compute it. Our formula for the Viterbo–Maslov
index in Theorem 3.4 and our combinatorial characterization of smooth lunes in
Theorem 6.7 are not needed for their applications. In our memoir these two results
are limited to the elements of D. (To our knowledge, they have not been extended
to triangles or more general polygons in the existing literature.)
When Σ =
T2,
the Heegaard–Floer theory of Ozsvath–Szabo [26, 27] can be
interpreted as a refinement of the combinatorial Floer theory, in that the wind-
ing number of a lune at a prescribed point in T2 \ β) is taken into account
in the definition of their boundary operator. However, for higher genus surfaces
Heegaard–Floer theory does not include the combinatorial Floer theory discussed
in the present memoir as a special case.
Appendix A contains a proof that, under suitable hypotheses, the space of
paths connecting α to β is simply connected. Appendix B contains a proof that the
group of orientation preserving diffeomorphisms of the half disc fixing the corners
is connected. Appendix C contains an account of Floer’s algebraic deformation ar-
gument. Appendix D summarizes the relevant results in [32] about the asymptotic
behavior of holomorphic strips.
Throughout this memoir we assume (H). We often write “assume (H)”
to remind the reader of our standing hypothesis.
Acknowledgement. We would like to thank the referee for his/her careful
work.
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