The Picard-Lefschetz theory of complexified Morse functions 1 ...

Complexified Morse functions 13
determined to a large degree by a canonical (up to isotopy) parameterization of ∂∆j,
as in [S08A, §16b].)
A third point of interest is to compare the Donalson and Seidel decompositions. Conjecturally,
the mapping cone of a morphism between two Lagrangians L1, L2, let’s say
corresponding to a single point in L1 ∩ L2, say α ∈ CF(L1, L2), is isomorphic to the
Lagrangian surgery of L1 and L2, say L1#L2:
Cone(α: L1 → L2) ∼ = L1#L2,
and a version of this is known if L1 is a Lagrangian sphere, see [S08A, §17j]. It would
be interesting to prove that Nj+1 = Nj#∆j+1 is isomorphic to Cone(Nj → ∆j+1) in
the above matching path construction. (There is a corresponding result for the case
of standard matching paths, [S08A, lemma 18.20].) This would show that whenever
we have a generalized matching path with corresponding Lagrangian L, there is a
Donaldson type decomposition of L which coincides with a Seidel decomposition of
L. Using this together with [AMP05], for example, one might be able to prove a new
version of Seidel’s decomposition (∗). This version would rely on choosing different
Lefschetz fibrations for different Lagrangians, rather than having one fixed Lefschetz
fibration.
1.3.3 Lagrangian submanifolds in T ∗ N
Here we elaborate a little on how Theorem A is relevant for the study of exact Lagrangian
submanifolds L ⊂ T ∗ N (see also the introduction to [J09A]). Our basic goal is
to prove for certain N that any closed exact Lagrangian submanifold L ⊂ T ∗ N is Floer
theoretically equivalent to N . This means in particular that HF(L, L) ∼ = HF(N, N), so
N), so that deg(L −→ N) = ±1.
that H∗ (L) ∼ = H∗ (N), and HF(L, T∗ x N) ∼ = HF(N, T∗ x
Of course, results of this kind have been obtained for arbitrary manifolds N in
[FSS08, FSS07] and [N07, NZ07]. We want to consider a slightly different approach
along the lines of the quiver-theoretic approach for the case N = Sn in [S04]. This approach
avoids spectral sequences and the use of gradings; thus it avoids one significant
assumption on L, namely that it has vanishing Maslov class µL ∈ H1 (L).
To keep things concrete, take N = CP 2 , and a Morse function f : N −→ R with
three critical points x0, x2, x4 , with Morse indices 0, 2, 4. Let (E,π) be the corresponding
Lefschetz fibration from Theorem A, which models the complexification of f on
D(T ∗ N). By construction, π comes with an explicit regular fiber M and vanishing
spheres L0, L2, L4 ⊂ M. The main consequence of Theorem A is that we have an exact