Citation: Zachary Mekus, Jessica Cooley, Aaron George, Victoria Sabo, Morgan Strzegowski, Michelle Starz-Gaiano, Bradford E. Peercy. Effects of cell packing on chemoattractant distribution within a tissue[J]. AIMS Biophysics, 2018, 5(1): 1-21. doi: 10.3934/biophy.2018.1.1
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Consider a symplectic manifold
Problem. Find conditions under which
This generalizes the problems of showing that a given Hamiltonian diffeomorphism has a fixed point and that a given Lagrangian submanifold intersects its image under a Hamiltonian diffeomorphism. References for solutions to the general problem are provided in [20,22].
Example(translated points). As explained in [19,p. 97], translated points of the time-1-map of a contact isotopy starting at the identity are leafwise fixed points of the Hamiltonian lift of this map to the symplectization.
We denote
Nω:={isotropic leaves of N}. |
We call
1Such a structure is unique if it exists. In this case the symplectic quotient of
We denote by
The main result of [20] (Theorem 1.1) implies the following. We denote by
Theorem 1.1 (leafwise fixed points for adiscal coisotropic). Assume that
2This means compact and without boundary.
|Fix(ψ,N)|≥dimN∑i=0bi(N). | (1) |
This bound is sharp if there exists a
3[20,Theorem 1.1] is formulated in a more general setting than Theorem 1.1. Chekanov's result is needed to deal with that setting, whereas in the setting of Theorem 1.1 Floer's original article [5] suffices.
Similarly to Theorem 1.1, in [21] for a regular
Theorem 1.2 (leafwise fixed points for monotone coisotropic). Assume that
4[20,Theorem 1.1] is stated for the geometrically bounded case, but the proof goes through in the convex at infinity case.
|Fix(ψ,N)|≥m(N)−2∑i=dimN−m(N)+2bi(N). | (2) |
The idea of the proof of this theorem given in [21], is to use the same Lagrangian embedding as in the proof of Theorem 1.1. We then apply P. Albers' Main Theorem in [2], which states Theorem 1.2 in the Lagrangian case.
Finally, the main result of [22] (Theorem 1) implies that leafwise fixed points exist for an arbitrary closed coisotropic submanifold if the Hamiltonian flow is suitably
Theorem 1.3 (leafwise fixed points for
|Fix(φ1,N)|≥dimN∑i=0bi(N). | (3) |
This result is optimal in the sense that the
The point of this note is to reinterpret the proofs of Theorems 1.1 and 1.2 in terms of a version of Floer homology for an adiscal or monotone regular coisotropic submanifold. I also outline a definition of a local version of Floer homology for an arbitrary closed coisotropic submanifold and use it to reinterpret the proof of Theorem 1.3. Details of the construction of this homology will be carried out elsewhere. For the extreme cases
5In [1] a Lagrangian Floer homology was constructed that is "local" in a different sense.
Potentially a (more) global version of coisotropic Floer homology may be defined under a suitable condition on
6This can only work under suitable conditions on
Based on the ideas outlined below, one can define a Floer homology for certain regular contact manifolds and use it to show that a given time-1-map of a contact isotopy has translated points. Namely, consider a closed manifold
Various versions of coisotropic Floer homology may play a role in mirror symmetry, as physicists have realized that the Fukaya category should be enlarged by coisotropic submanifolds, in order to make homological mirror symmetry work, see e.g. [11].
To explain the coisotropic Floer homology in the regular case, consider a geometrically bounded symplectic manifold
Suppose first also that
7 By definition, for every such point
Fixc(N,φ):={(N,φ)-contractible leafwise fixed points},CF(N,φ):=⊕Fixc(N,φ)Z2. | (4) |
Remark. By definition this direct sum contains one copy of
We now define a collection of boundary operators on
ˆM:=M×Nω,ˆω:=ω⊕(−ωN),ιN:N→ˆM,ιN(x):=(x,isotropic leaf through x),ˆN:=ιN(N),ˆφt:=φt×idNω. | (5) |
The map
ιN:Fixc(N,φ)→Fixc(ˆN,ˆφ)={ˆx∈ˆN∩(ˆφ1)−1(ˆN)|t↦ˆφt(ˆx) contractible with endpoints in ˆN} | (6) |
is well-defined and injective. A straightforward argument shows that it is surjective.
Let
8 The exponent
9 It follows from the proof of [5,Proposition 2.1] that this set is dense in the set of all
∂N,φ,ˆJ:CF(N,φ)→CF(N,φ) |
to be the (Lagrangian) Floer boundary operator of
To see that this operator is well-defined, recall that it is defined on the direct sum of
10 Sometimes this is called the "
We check the conditions of [5,Definition 3.1]. Since
11In [5] Floer assumes that the symplectic manifold is closed. However, the same construction of Floer homology works for geometrically bounded symplectic manifolds. Here we use that we only consider Floer strips with compact image.
HF(N,φ,ˆJ):=H(CF(N,φ),∂N,φ,ˆJ). |
Let
12By [5,Proposition 2.4] such a grading exists and each two gradings differ by an additive constant.
Φ^J0,^J1:HF(N,φ,^J0)→HF(N,φ,^J1) |
the canonical isomorphism provided by the proof of [5,Proposition 3.1,p. 522]. This isomorphism respects the grading
Definition 2.1 (Floer homology for adiscal coisotropic). We define the Floer homology of
HF(N,φ):=((HF(N,φ,ˆJ))ˆJ∈Jreg(N,φ1),(Φ^J0,^J1)^J0,^J1∈Jreg(N,φ1)). |
Remarks. ● This is a collection of graded
● Philosophically, the Floer homology of
By the proof of [5,Theorem 1]
Suppose now that
13We continue to assume that
Definition 2.2 (Floer homology for monotone coisotropic). We define the Floer homology of
Since
Consider now the situation in which
To explain the boundary operator
˜N:={(x,x)|x∈N} | (7) |
as a Lagrangian submanifold. We shrink
The boundary operator
To understand why heuristically, the boundary operator
14Here one needs to work with a family of almost complex structures depending on the time
● Holomorphic strips with boundary on
● Disks or spheres cannot bubble off. This follows from our assumption that
● Index-1-strips generically do not break.
It follows that heuristically,
Given two choices of symplectic submanifolds
To make the outlined Floer homology rigorous, the words "close" and "short" used above, need to be made precise. To obtain an object that does not depend on the choice of "closeness", the local Floer homology of
φ↦HF(N,φ,J) |
around
By showing that
Remark(local presymplectic Floer homology). A presymplectic form on a manifold is a closed two-form with constant rank. By [12,Proposition 3.2] every presymplectic manifold can be coisotropically embedded into some symplectic manifold. By [12,4.5. Théorème on p. 79] each two coisotropic embeddings are equivalent. Hence heuristically, we may define the local Floer homology of a presymplectic manifold to be the local Floer homology of any of its coisotropic embeddings.
Remark (relation between the constructions). Assume that
(x,y)↦(x,isotropic leaf through y). |
I would like to thank Will Merry for an interesting discussion and the anonymous referees for valuable suggestions.
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