The relative Maslov grading of generators #
Let (Σ, α, β, z) be a pointed Heegaard diagram with n attaching curves of each kind and a
single basepoint z. The Heegaard Floer chain group CF̂ is relatively graded: for generators
x, y joined by a domain D, the relative Maslov grading is
gr(x, y) = μ(D) - 2 n_z(D),
where μ is the combinatorial Maslov index TauCeti.HeegaardRegionSystem.maslovIndex and
n_z(D) is the multiplicity of D at the basepoint. The hat differential counts holomorphic
disks of Maslov index one avoiding the basepoint, so it lowers gr by one.
Two domains from x to y differ by a domain from x to x. With a single basepoint, such a
domain is a periodic domain plus a multiple of the whole surface Σ, the domain 1. The whole
surface has n_z(Σ) = 1 and, when the Euler characteristics χ of the regions are those of the
regions of a closed surface of genus n, Maslov index μ(Σ) = 2. So it does not change gr,
while a periodic domain P changes it by μ(P). Hence gr is well defined as soon as every
periodic domain has Maslov index zero at x. On a geometric diagram μ(P) is the evaluation of
the first Chern class of the spin^c structure of x on the homology class of P, so this holds
for the generators of a torsion spin^c structure, and in particular when b₁(Y) = 0, when the
only periodic domain is zero. Without it gr genuinely depends on the domain: for the
genus-one diagram of S¹ × S² with its basepoint in a bigon, the two bigons from p₀ to p₁
give the values -1 and 1
(TauCeti.HeegaardRegionSystem.maslovIndex_sub_two_mul_circleTimesSphere_one).
In general gr is a relative grading modulo the Maslov indices of the periodic domains at x,
the subgroup TauCeti.HeegaardRegionSystem.periodicMaslovIndices of ℚ. On a geometric diagram,
by the evaluation of the first Chern class recalled above (not formalized here), this is the
image of the pairing of c₁(s_z(x)) with H₂(Y; ℤ), so it is d(s) ℤ for the divisibility
d(s) of c₁(s_z(x)), and gr is the relative ℤ / d(s) grading of Ozsváth and Szabó.
Combinatorially the subgroup is cyclic, generated by some d ≥ 0, because every Maslov index is
a multiple of 1 / 4. It is the same at generators joined by a domain, and the
class of μ(D) - 2 n_z(D) in the quotient of ℚ by it does not depend on the domain D from
x to y. The value TauCeti.HeegaardRegionSystem.relativeGrading computed from a chosen domain
is therefore a representative of this class, additive and antisymmetric modulo the subgroup, and
the hat differential lowers it by one modulo the subgroup. The ℚ-valued statements above are the
case where the subgroup is zero.
The Euler characteristics are not determined by the incidence data, so they are an explicit
argument χ, normalized by ∑ χ(R) - #(α ∩ β) = 2 - 2n. The left side is the Euler
characteristic of the surface assembled from the regions, the 2 #(α ∩ β) arcs and the
intersection points, so this says that the surface has genus n, as for a pointed Heegaard
diagram.
Main definitions #
TauCeti.HeegaardRegionSystem.periodicMaslovIndices: the subgroup ofℚof Maslov indices of periodic domains at a generator.TauCeti.HeegaardRegionSystem.relativeGrading: the relative gradinggr(x, y), computed from a chosen domain fromxtoy, or0when there is none.
Main results #
TauCeti.HeegaardRegionSystem.exists_nonneg_periodicMaslovIndices_eq_zmultiples: the subgroup of periodic Maslov indices is generated by somed ≥ 0.TauCeti.HeegaardRegionSystem.IsDomainBetween.periodicMaslovIndices_eq: it is the same at generators joined by a domain.TauCeti.HeegaardRegionSystem.IsDomainBetween.coe_relativeGrading_eq: modulo the periodic Maslov indices, the relative grading isμ(D) - 2 n_z(D)for every domainDfromxtoy.TauCeti.HeegaardRegionSystem.IsDomainBetween.coe_relativeGrading_addandTauCeti.HeegaardRegionSystem.IsDomainBetween.coe_relativeGrading_swap: additivity and antisymmetry modulo the periodic Maslov indices.TauCeti.HeegaardRegionSystem.IsDomainBetween.maslovIndex_self_eq_two_mul: a domain fromxtoxhas Maslov index2 n_z.TauCeti.HeegaardRegionSystem.IsDomainBetween.maslovIndex_sub_two_mul_eq:μ(D) - 2 n_z(D)is the same for all domainsDfromxtoy.TauCeti.HeegaardRegionSystem.IsDomainBetween.relativeGrading_eq: the relative grading isμ(D) - 2 n_z(D)for every domainDfromxtoy.TauCeti.HeegaardRegionSystem.IsDomainBetween.relativeGrading_add:gr(x, y) + gr(y, w) = gr(x, w).TauCeti.HeegaardRegionSystem.IsDomainBetween.relativeGrading_swap:gr(y, x) = -gr(x, y).
References #
- P. Ozsváth and Z. Szabó, Holomorphic disks and topological invariants for closed
three-manifolds, Ann. of Math. 159 (2004),
arXiv:math/0101206: the relative grading
gr(x, y) = μ(φ) - 2 n_z(φ), the class of the whole surface with Maslov index two and basepoint multiplicity one, and the evaluation of the first Chern class as the Maslov index of a periodic domain. - R. Lipshitz, A cylindrical reformulation of Heegaard Floer homology, Geom. Topol. 10
(2006), arXiv:math/0502404, §4, for the index formula
μ(D) = e(D) + n_x(D) + n_y(D).
The Maslov indices at the generator x of the periodic domains, a subgroup of ℚ. In a
singly pointed diagram the relative grading is well defined modulo this subgroup
(TauCeti.HeegaardRegionSystem.IsDomainBetween.coe_relativeGrading_eq). On a geometric diagram,
by Ozsváth and Szabó (not formalized here), it is the image of the pairing of c₁(s_z(x)) with
H₂(Y; ℤ).
Instances For
The periodic Maslov indices are the image of the periodic domains under the Maslov index.
A rational number is a periodic Maslov index at x when it is the Maslov index at x of a
periodic domain.
The Maslov index at x of a periodic domain is a periodic Maslov index at x.
The periodic Maslov indices at x vanish exactly when every periodic domain has Maslov
index zero at x, the hypothesis of the ℚ-valued relative grading.
Generators joined by a domain have the same periodic Maslov indices.
The periodic Maslov indices at x are the integer multiples of some d ≥ 0, the
combinatorial counterpart of the divisibility d(s) of c₁(s). The subgroup is cyclic because
every Maslov index is an integer multiple of 1 / 4.
The relative Maslov grading gr(x, y) = μ(D) - 2 n_z(D) of two generators, for a chosen
domain D from x to y, or 0 when no domain joins them. In a singly pointed diagram
satisfying the Euler normalization, its class modulo the periodic Maslov indices at x does not
depend on the choice of D (TauCeti.HeegaardRegionSystem.IsDomainBetween.coe_relativeGrading_eq),
and when the periodic domains have Maslov index zero at the source generator neither does the
value itself (TauCeti.HeegaardRegionSystem.IsDomainBetween.relativeGrading_eq).
Instances For
In a singly pointed diagram of a closed surface of genus n, a domain E from x to x
has μ(E) - 2 n_z(E) among the periodic Maslov indices at x: it is a periodic domain plus
n_z(E) copies of the whole surface, which has Maslov index two.
In a singly pointed diagram of a closed surface of genus n all of whose periodic domains
have Maslov index zero at x, a domain E from x to x has Maslov index 2 n_z(E): it is a
periodic domain plus n_z(E) copies of the whole surface.
In a singly pointed diagram of a closed surface of genus n, the quantities
μ(D) - 2 n_z(D) for domains D from x to y differ by periodic Maslov indices at x.
In a singly pointed diagram of a closed surface of genus n all of whose periodic domains
have Maslov index zero at x, the quantity μ(D) - 2 n_z(D) is the same for all domains D
from x to y.
When no domain joins x to y, the relative grading is 0.
Modulo the periodic Maslov indices at x, the relative grading is μ(D) - 2 n_z(D) for
every domain D from x to y. This is the combinatorial counterpart of the relative
ℤ / d(s) grading of Ozsváth and Szabó; the identification with c₁ is not formalized here.
The relative grading is μ(D) - 2 n_z(D) for every domain D from x to y, when the
periodic domains have Maslov index zero at x.
A generator has relative grading zero with itself, modulo the periodic Maslov indices.
The relative grading is additive modulo the periodic Maslov indices:
gr(x, y) + gr(y, w) = gr(x, w) for generators joined by domains.
The relative grading is antisymmetric modulo the periodic Maslov indices:
gr(y, x) = -gr(x, y) for generators joined by a domain.
A generator has relative grading zero with itself.
The relative grading is additive: gr(x, y) + gr(y, w) = gr(x, w) for generators joined by
domains, when the periodic domains have Maslov index zero at x.
The relative grading is antisymmetric: gr(y, x) = -gr(x, y) for generators joined by a
domain, when the periodic domains have Maslov index zero at x.