The Floer homology of a transverse pair of Lagrangian submanifolds in a sym-
plectic manifold is, under favorable hypotheses, the homology of a chain complex
generated by the intersection points. The boundary operator counts index one
holomorphic strips with boundary on the Lagrangian submanifolds. This theory
was introduced by Floer in [10, 11]; see also the three papers [21] of Oh. In this
memoir we consider the following special case:
(H) Σ is a connected oriented 2-manifold without boundary and
α, β Σ are connected smooth one dimensional oriented sub-
manifolds without boundary which are closed as subsets of Σ and
intersect transversally. We do not assume that Σ is compact,
but when it is, α and β are embedded circles.
In this special case there is a purely combinatorial approach to Lagrangian Floer
homology which was first developed by de Silva [6]. We give a full and detailed
definition of this combinatorial Floer homology (see Theorem 9.1) under the hy-
pothesis that α and β are noncontractible embedded circles and are not isotopic
to each other. Under this hypothesis, combinatorial Floer homology is invariant
under isotopy, not just Hamiltonian isotopy, as in Floer’s original work (see Theo-
rem 9.2). Combinatorial Floer homology is isomorphic to analytic Floer homology
as defined by Floer (see Theorem 9.3).
Floer homology is the homology of a chain complex CF(α, β) with basis consist-
ing of the points of the intersection α β (and coefficients in Z/2Z). The boundary
operator : CF(α, β) CF(α, β) has the form
∂x =
n(x, y)y.
In the case of analytic Floer homology as defined by Floer n(x, y) denotes the
number (mod two) of equivalence classes of holomorphic strips v : S Σ satisfying
the boundary conditions
v(R) α, v(R + i) β, v(−∞) = x, v(+∞) = y
and having Maslov index one. The boundary operator in combinatorial Floer ho-
mology has the same form but now n(x, y) denotes the number (mod two) of equiv-
alence classes of smooth immersions u : D Σ satisfying
u(D R) α, u(D
β, u(−1) = x, u(+1) = y.
We call such an immersion a smooth lune. Here
S := R + i[0, 1], D := {z C | Im z 0, |z| 1}
denote the standard strip and the standard half disc respectively. We develop
the combinatorial theory without appeal to the difficult analysis required for the
Previous Page Next Page