analytic theory. The invariance under isotopy rather than just Hamiltonian isotopy
(Theorem 9.3) is a benefit of this approach. A corollary is the formula
dim HF(α, β) = geo (α, β)
for the dimension of the Floer Homology HF(α, β) (see Corollary 9.5). Here
geo (α, β) denotes the geometric intersection number of the curves α and β. In
Remark 9.11 we indicate how to define combinatorial Floer homology with integer
coefficients, but we do not discuss integer coefficients in analytic Floer homology.
Let D denote the space of all smooth maps u : D Σ satisfying the boundary
conditions u(D R) α and u(D
β. For x, y α β let D(x, y) denote
the subset of all u D satisfying the endpoint conditions u(−1) = x and u(1) = y.
Each u D determines a locally constant function
w : Σ \ β) Z
defined as the degree
w(z) := deg(u, z), z Σ \ β).
When z is a regular value of u this is the algebraic number of points in the preimage
u−1(z). The function w depends only on the homotopy class of u. In Theorem 2.4
we prove that the homotopy class of u D is uniquely determined by its endpoints
x, y and its degree function w. Theorem 3.4 says that the Viterbo–Maslov index of
every smooth map u D(x, y) is determined by the values of w near the endpoints
x and y of u, namely, it is given by the following trace formula
μ(u) =
mx(Λu) + my(Λu)
, Λu := (x, y, w).
Here mx denotes the sum of the four values of w encountered when walking along
a small circle surrounding x, and similarly for y. Part I of this memoir is devoted
to proving this formula.
Part II gives a combinatorial characterization of smooth lunes. Specifically, the
equivalent conditions (ii) and (iii) of Theorem 6.7 are necessary for the existence
of a smooth lune. This implies the fact (not obvious to us) that a lune cannot
contain either of its endpoints in the interior of its image. In the simply connected
case we prove in the same theorem that the necessary conditions are also sufficient.
We conjecture that they characterize smooth lunes in general. Theorem 6.8 shows
that any two smooth lunes with the same counting function w are isotopic and thus
the equivalence class of a smooth lune is uniquely determined by its combinatorial
data. The proofs of these theorems are carried out in Chapters 7 and 8. Together
they provide a solution to the Picard–Loewner problem in a special case; see for
example [12] and the references cited therein, e.g. [4,28,38]. Our result is a special
case because no critical points are allowed (lunes are immersions), the source is a
disc and not a Riemann surface with positive genus, and the prescribed boundary
circle decomposes into two embedded arcs.
Part III introduces combinatorial Floer homology. Here we restrict our dis-
cussion to the case where α and β are noncontractible embedded circles which
are not isotopic to each other (with either orientation). The basic definitions are
given in Chapter 9. That the square of the boundary operator is indeed zero in
the combinatorial setting will be proved in Chapter 10 by analyzing broken hearts.
Propositions 10.2 and 10.5 say that there are two ways to break a heart and this
is why the square of the boundary operator is zero. In Chapter 11 we prove the
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