Caps of contractible orbits as torsors over the period group #
Let x be a contractible one-periodic orbit of a Hamiltonian on a closed symplectic manifold
(X, ω). A raw cap of x is a homotopy class of discs bounding x. Gluing a sphere to a disc
makes the raw caps of x a torsor P under a group S of spherical classes on which the
symplectic area ω : S →+ ℝ and the first Chern number c : S →+ ℤ are defined (for instance
π₂(X) or its Hurewicz image). Capped Floer theory identifies two caps when the sphere obtained
by gluing them along x has zero area and zero first Chern number. The resulting cap classes
CapClass ω c P form a torsor under the period group Γ = PeriodGroup ω c; this is an instance
of the general fact that H-orbits in a G-torsor form a G ⧸ H-torsor
(AddAction.orbitRel.Quotient.instAddTorsorQuotientAddGroup).
The action A_H(x, D) = ∫ D^* ω + ∫ H(t, x(t)) dt and the grading μ(x, D) of a capped orbit
obey the recapping laws A_H(x, B # D) = ω(B) + A_H(x, D) and μ(x, B # D) = 2 c₁(B) + μ(x, D).
A function on raw caps obeying such a law descends to cap classes (CapClass.lift), and a cap
class is determined by its action and its grading (eq_iff_action_eq_and_index_eq): this is
why the period group is the quotient by the joint kernel of area and Chern number.
For the pair-of-pants product, fix a homotopy class u of pairs of pants with input orbits
x₁, x₂ and output orbit x₀. Gluing caps D₁, D₂ of the inputs to u gives a cap glue D₁ D₂
of the output, and recapping either input recaps the glued cap by the same class. The class
A = D₀ + u − D₁ − D₂ of the resulting product coefficient, measured against an output cap D₀,
is productClass glue D₁ D₂ D₀ = D₀ -ᵥ glue D₁ D₂ ∈ Γ. Recapping an input by B changes A by
−B, and recapping the output changes it by +B. Consequently the topological energy
A₁(x₁, D₁) + A₂(x₂, D₂) − A₀(x₀, D₀) + ω(A) and the index
μ₁ + μ₂ − μ₀ + 2 c₁(A) do not depend on the choice of caps
(add_sub_add_map_productClass_eq, add_sub_add_two_mul_chern_productClass_eq).
Main definitions #
TauCeti.Floer.CapClass ω c P: cap classes, the quotient of the raw capsPby recapping with classes of zero area and zero first Chern number. It is a torsor underPeriodGroup ω c.TauCeti.Floer.CapClass.lift: the descent of a function on raw caps obeying a recapping law.TauCeti.Floer.productClass: the classD₀ -ᵥ glue D₁ D₂of a product coefficient.
Main results #
TauCeti.Floer.CapClass.mk_eq_mk_iff: two raw caps have the same class exactly when they differ by a class of zero area and zero first Chern number, equivalently (CapClass.mk_eq_mk_iff_action_eq_and_index_eq) when they have the same action and grading.TauCeti.Floer.CapClass.lift_vadd: the descended function obeys the recapping law on cap classes.TauCeti.Floer.eq_iff_action_eq_and_index_eq: a cap class is determined by its action and its grading.TauCeti.Floer.productClass_vadd_fst,TauCeti.Floer.productClass_vadd_snd,TauCeti.Floer.productClass_vadd_out: the recapping laws for the product class.TauCeti.Floer.productClass_eq_add_sub: the change of reference caps for the product class.TauCeti.Floer.add_sub_add_map_productClass_eq: the cap independence off₁ + f₂ − f₀ + φ(A)for functionsfᵢobeying the recapping law forφ; withφthe area this is the cap independence of the topological energy.TauCeti.Floer.add_sub_add_two_mul_chern_productClass_eq: the cap independence of the indexμ₁ + μ₂ − μ₀ + 2 c₁(A).
References #
- H. Hofer and D. A. Salamon, Floer homology and Novikov rings, in The Floer memorial volume, Progr. Math. 133 (1995), 483–524.
- S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer–Donaldson theory and quantum cohomology, in Contact and symplectic geometry, Publ. Newton Inst. 8 (1996), 171–200.
- D. McDuff and D. Salamon, J-holomorphic curves and symplectic topology, the chapter on Floer homology and the pair-of-pants product.
The cap classes of an orbit: the quotient of a torsor P of raw caps under the spherical
classes S by recapping with classes of zero area and zero first Chern number. It is a torsor
under the period group PeriodGroup ω c, and recapping by γ : PeriodGroup ω c is γ +ᵥ D.
It is an abbreviation so that the orbit-quotient and torsor API applies to it directly.
Instances For
A function f on raw caps that changes by φ g under recapping by g, for a homomorphism
φ on the period group, descends to cap classes. The action (with φ = PeriodGroup.area ω c)
and the grading (with φ = 2 • PeriodGroup.chern ω c) are the main examples.
Instances For
The descended function takes the value of f on the class of a raw cap.
The descended function obeys the recapping law on cap classes.
Caps are determined by their action and their grading: in a torsor C under the period
group, such as CapClass ω c P, if a and m obey the recapping laws for area and twice the
first Chern number, then two caps are equal exactly when they have the same value under both.
Two raw caps have the same class exactly when they have the same action and the same grading,
for functions a and m on raw caps obeying the recapping laws for area and twice the first
Chern number.
The class A = D₀ + u − D₁ − D₂ of a product coefficient. Here glue D₁ D₂ is the cap of
the output orbit obtained by gluing input caps D₁ and D₂ to a fixed pair of pants u, and
D₀ is the chosen output cap. In a reference-output basis the coefficient is weighted by the
Novikov monomial of -A.
Instances For
The product class vanishes exactly when the output cap is the glued cap.
Recapping the output by γ adds γ to the product class.
If gluing is equivariant in the first input, recapping the first input by γ subtracts γ
from the product class.
If gluing is equivariant in the second input, recapping the second input by γ subtracts
γ from the product class.
Change of reference caps: if gluing is equivariant in each input, the product class for caps
Dᵢ is the product class for reference caps rᵢ, plus the recapping class of the output, minus
the recapping classes of the inputs.
Cap independence of f₁ D₁ + f₂ D₂ − f₀ D₀ + φ(A): if gluing is equivariant in each input
and the functions fᵢ change by φ γ under recapping by γ, then this quantity takes the same
value for all choices of caps. With φ = PeriodGroup.area ω c and the action functionals as
fᵢ, this is the cap independence of the topological energy
A₁(D₁) + A₂(D₂) − A₀(D₀) + ω(A) of a product coefficient.
Cap independence of the index μ₁(D₁) + μ₂(D₂) − μ₀(D₀) + 2 c₁(A) of a product coefficient,
for gradings mᵢ obeying the recapping law for twice the first Chern number. The Fredholm index
of a pair of pants in class A is this quantity minus half the dimension of the target.