Bilinear products of completed combinations given by supported kernels #
Let F₁ : S₁ → ℝ, F₂ : S₂ → ℝ and F₀ : S₀ → ℝ be potentials, and let
n : S₁ → S₂ → S₀ → R be a product kernel: n p₁ p₂ p₀ is the coefficient of the output
generator p₀ in the product of the input generators p₁ and p₂. Suppose that
n p₁ p₂ p₀ ≠ 0impliesF₀ p₀ ≤ F₁ p₁ + F₂ p₂ + Cfor a fixed constantC, and- each output function
n p₁ p₂is itself a completed combination, that is, it has only finitely many nonzero coefficients above every threshold ofF₀.
Then for completed combinations a and b the sum ∑ a(p₁) b(p₂) n(p₁, p₂, p₀) has finitely many
nonzero terms, and it defines a bilinear map
NovikovModule R F₁ → NovikovModule R F₂ → NovikovModule R F₀ (NovikovModule.kernelProduct). The
product shifts potential cutoffs by C: if a and b vanish above D₁ and D₂, then their
product vanishes above D₁ + D₂ + C.
This is the algebraic form of the pair-of-pants product on capped Hamiltonian Floer complexes.
The generators are capped orbits, the potentials are the action functionals, n counts rigid
pairs of pants whose glued cap is the output cap, and the potential bound is the energy estimate
0 ≤ E = A₁ + A₂ - A₀ - ∫ R_K, with C bounding -∫ R_K for the curvature R_K of the
perturbation. Finiteness of each n p₁ p₂ above an action level is compactness of the moduli
spaces below an energy level.
Suppose next that each Sₗ is a finite disjoint union of torsors under a group H, that the
potentials satisfy F (h +ᵥ p) = F p - N h, and that the kernel is unchanged when one input and
the output are translated by the same element (recapping both by the same sphere). Choosing one
reference generator in each orbit turns the kernel into a finite matrix of Novikov series
(NovikovModule.kernelMatrix). The (i, j, k) entry lies in the period filtration at level
F₀(r₀ k) - F₁(r₁ i) - F₂(r₂ j) - C; this is the lower valuation bound of the output monomials.
In the lift coordinates of TauCeti.RingTheory.NovikovModule.Basic the product is given by the
matrix formula (a ⋆ b)ₖ = ∑ᵢ ∑ⱼ aᵢ bⱼ Mᵢⱼₖ (NovikovModule.coordinates_kernelProduct), even
for a noncommutative group H. Fixing one input turns the product into a one-input invariant
kernel map of TauCeti.RingTheory.NovikovModule.KernelMap, whose coordinates are given by its own
matrix (NovikovModule.coordinates_eq_sum_mul). Hence the product is linear over the Novikov ring
in each argument, using only the invariance and equivariance on that side, and for a commutative
group it is a Novikov-bilinear map (NovikovModule.novikovKernelProduct). No discreteness or
injectivity of the periods N is assumed.
Conversely, for a commutative group H every matrix M of Novikov series is the matrix of an
invariant kernel (NovikovModule.kernelOfMatrix): the coefficient on generators in the orbits
i, j and k is the coefficient of M i j k at the displacement of the output from its
reference generator minus the displacements of the two inputs. The two constructions are inverse
to each other (NovikovModule.kernelMatrix_kernelOfMatrix,
NovikovModule.kernelOfMatrix_kernelMatrix). The kernel of a matrix whose entries obey the lower
period bounds above obeys the potential bound with the same constant, and with finitely many
orbits the kernel of every matrix obeys a potential bound for some constant. Hence every finite
matrix of Novikov series has a Novikov-bilinear product (NovikovModule.novikovProductOfMatrix),
given in reference coordinates by the matrix formula, and every Novikov-bilinear map of these
finitely generated modules is the product of the matrix of its values on the reference basis
(NovikovModule.eq_novikovProductOfMatrix). In Floer theory, the entries are the count series
∑_A #M(x₁, x₂, x₀; A) e^(-A) of the reference generator triples, with A measured against the
reference caps, and the product of this matrix is the pair-of-pants product on the completed
complexes.
Composing a product with kernel maps, on either input or on the output, gives the product of the
composite kernel (NovikovModule.kernelMap_kernelProduct,
NovikovModule.kernelProduct_kernelMap_left, NovikovModule.kernelProduct_kernelMap_right), and
the product is linear in the kernel (NovikovModule.kernelProduct_add,
NovikovModule.kernelProduct_smul). Hence a Leibniz rule
d (a ⋆ b) = d a ⋆ b ± a ⋆ d b for differentials given by kernels follows from the corresponding
identity between kernels, which is what counting the ends of one-dimensional moduli spaces gives.
Main definitions #
TauCeti.NovikovModule.kernelProduct: the bilinear product given by a supported kernel.TauCeti.NovikovModule.toProductKernel: the coefficients of a bilinear map.TauCeti.NovikovModule.kernelMatrix: the matrix of Novikov series of an invariant kernel in chosen reference generators.TauCeti.NovikovModule.novikovKernelProduct: the product of an invariant kernel, as a bilinear map over the Novikov ring.TauCeti.NovikovModule.kernelOfMatrix: the invariant kernel with a given matrix of Novikov series.TauCeti.NovikovModule.novikovProductOfMatrix: the Novikov-bilinear product of a finite matrix of Novikov series.
Main results #
TauCeti.NovikovModule.kernelProduct_apply_val_eq_zero: the product shifts potential cutoffs by the constant of the potential bound.TauCeti.NovikovModule.kernelMatrix_mem_periodFiltration: the uniform lower period bound on the matrix entries.TauCeti.NovikovModule.coordinates_kernelProduct: the matrix formula for the product in lift coordinates.TauCeti.NovikovModule.kernelProduct_eq_kernelMap_left,TauCeti.NovikovModule.kernelProduct_eq_kernelMap_right: for one input fixed, the product is the map of a one-input kernel.TauCeti.NovikovModule.kernelMap_kernelProduct,TauCeti.NovikovModule.kernelProduct_kernelMap_left,TauCeti.NovikovModule.kernelProduct_kernelMap_right: composites of products and kernel maps are products of composite kernels.TauCeti.NovikovModule.kernelProduct_smul_left,TauCeti.NovikovModule.kernelProduct_smul_right: linearity over the Novikov ring.TauCeti.NovikovModule.kernelMatrix_kernelOfMatrix,TauCeti.NovikovModule.kernelOfMatrix_kernelMatrix: invariant kernels and matrices of Novikov series determine each other.TauCeti.NovikovModule.exists_potential_le_of_kernelOfMatrix_ne_zero,TauCeti.NovikovModule.potential_le_of_kernelOfMatrix_ne_zero: potential bounds for the kernel of a matrix.TauCeti.NovikovModule.coordinates_novikovProductOfMatrix: the matrix formula for the product of a matrix.TauCeti.NovikovModule.eq_novikovProductOfMatrix: every Novikov-bilinear map is the product of a matrix.
References #
- M. Schwarz, Cohomology operations from S¹-cobordisms in Floer homology, PhD thesis, ETH Zürich (1995), on the pair-of-pants product.
- 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.
- M. Hutchings and Y.-J. Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999), Section 2, on completed modules over Novikov rings.
Only finitely many pairs of support generators of two completed combinations have total potential at least a given threshold.
For a kernel obeying a potential bound, each coefficient of the product is a sum with finitely many nonzero terms.
The bilinear product of completed combinations given by a kernel n with potential bound C
and completed outputs: the coefficient of p₀ in the product of a and b is the finite sum
∑ a(p₁) b(p₂) n(p₁, p₂, p₀) over all pairs of generators.
Instances For
The coefficient of a generator in a product is the sum, over all pairs of input generators, of the product of their coefficients with the kernel.
A coefficient of a product is a sum over any finite set of pairs containing every pair of support generators with nonzero kernel coefficient.
The product shifts potential cutoffs by the constant of the potential bound: if a vanishes
above D₁ and b vanishes above D₂, then their product vanishes above D₁ + D₂ + C.
A kernel product sends a pair of generators to the corresponding entry of the kernel.
A product kernel whose outputs have finitely many nonzero coefficients maps pairs of finite combinations to finite combinations.
The coefficients of a bilinear map of completed combinations: the coefficient of the output
generator p₀ in the image of the pair of input generators p₁ and p₂.
Instances For
Each output of the coefficients of a bilinear map is a completed combination.
The coefficients of a kernel product are its kernel.
The product of a sum of kernels is the sum of their products. The three potential bounds may have different constants.
The product of a scalar multiple of a kernel is the scalar multiple of its product. The two potential bounds may have different constants.
Fixing the second input b of a supported product kernel gives the one-input kernel
p₁ ↦ ∑ b(p₂) n(p₁, p₂, ·), which obeys a potential bound: its constant uses an upper bound of
the potential on the support of b.
Fixing the first input a of a supported product kernel gives the one-input kernel
p₂ ↦ ∑ a(p₁) n(p₁, p₂, ·), which obeys a potential bound: its constant uses an upper bound of
the potential on the support of a.
The one-input kernel obtained by fixing the second input of a supported product kernel has completed outputs.
The one-input kernel obtained by fixing the first input of a supported product kernel has completed outputs.
A product as a kernel map in its first input. For fixed b, the product of a and b
is the image of a under the one-input kernel p₁ ↦ ∑ b(p₂) n(p₁, p₂, ·), for any potential
bound of that kernel.
A product as a kernel map in its second input. For fixed a, the product of a and b
is the image of b under the one-input kernel p₂ ↦ ∑ a(p₁) n(p₁, p₂, ·), for any potential
bound of that kernel.
A kernel map applied to a product. Applying the map of a kernel m to the product of
a and b is the product of a and b for the kernel obtained by composing each output of n
with m. The potential bound of the composite may have any constant.
A product with a kernel map applied to its first input. The product of the image of a
under the map of a kernel m with b is the product of a and b for the kernel obtained by
composing m with n. The potential bound of the composite may have any constant.
A product with a kernel map applied to its second input. The product of a with the
image of b under the map of a kernel m is the product of a and b for the kernel obtained
by composing m with n. The potential bound of the composite may have any constant.
Translating both inputs of an invariant kernel is the same as translating its output by the negatives, in reverse order.
The matrix of a kernel relative to reference generators r₁, r₂ and r₀: the coefficient
of its (i, j, k) entry at h is the coefficient of the output generator h +ᵥ r₀ k in the
product of the reference inputs r₁ i and r₂ j.
Instances For
The coefficient at h of a kernel matrix entry is the kernel coefficient of the reference
inputs and the output generator h +ᵥ r₀ k.
Under a potential bound with constant C, the (i, j, k) entry of the kernel matrix lies in
the period filtration at level F₀(r₀ k) - F₁(r₁ i) - F₂(r₂ j) - C.
In lift coordinates, the product of an invariant kernel is given by its kernel matrix:
(a ⋆ b)ₖ = ∑ᵢ ∑ⱼ aᵢ bⱼ Mᵢⱼₖ.
The product of a kernel invariant in its first input is linear over the Novikov ring in its first argument.
The product of a kernel invariant in its second input is linear over the Novikov ring in its second argument.
The product of an invariant kernel, as a bilinear map over the Novikov ring, for the Novikov
module structures of TauCeti.NovikovModule.module.
Instances For
The Novikov-bilinear product evaluates to the product of the kernel.
The product kernel with a given matrix M of Novikov series relative to reference generators
r₁, r₂ and r₀. Its coefficient on generators p₁, p₂ and p₀ in the orbits i, j and
k is the coefficient of M i j k at the displacement of p₀ from r₀ k, minus the
displacements of p₁ and p₂ from r₁ i and r₂ j. On invariant kernels it is the inverse of
NovikovModule.kernelMatrix.
Instances For
The coefficient of the kernel of a matrix on generators in the orbits i, j and k is a
coefficient of the (i, j, k) entry.
The kernel of a matrix is unchanged when its first input and its output are translated by the same element.
The kernel of a matrix is unchanged when its second input and its output are translated by the same element.
Every finite matrix of Novikov series has a supported kernel. With finitely many orbits,
the kernel of any matrix of Novikov series obeys a potential bound with some constant C, since
the periods of the support of each of the finitely many entries are bounded below.
The outputs of the kernel of a matrix are completed combinations: in the output coordinates
relative to r₀, the output at p₁ and p₂ is the family M i j · e^(u + v), where u and v
are the displacements of p₁ and p₂ from their reference generators.
The matrix of the kernel of a matrix, relative to the same reference generators, is the matrix itself.
An invariant kernel is the kernel of its matrix: kernels invariant under translating either input together with the output are determined by their matrices.
The potential bound of the kernel of a matrix. If the (i, j, k) entry of the matrix lies
in the period filtration at level F₀(r₀ k) - F₁(r₁ i) - F₂(r₂ j) - C, then the kernel obeys the
potential bound with constant C. This is the converse of
NovikovModule.kernelMatrix_mem_periodFiltration.
The Novikov-bilinear product of a matrix of Novikov series. For finitely many orbits and
reference generators r₁, r₂ and r₀, every matrix M of Novikov series defines the product
of its kernel NovikovModule.kernelOfMatrix, a bilinear map over the Novikov ring for the module
structures of TauCeti.NovikovModule.module. In the reference coordinates it is
(a ⋆ b)ₖ = ∑ᵢ ∑ⱼ aᵢ bⱼ Mᵢⱼₖ (NovikovModule.coordinates_novikovProductOfMatrix).
Instances For
The coefficient of a generator in the product of a matrix is the finite sum, over all pairs of input generators, of the product of their coefficients with the kernel of the matrix.
The product of a matrix is the kernel product of its kernel, for any potential bound of that
kernel. Thus the results on kernel products, such as the shift of potential cutoffs by C and
continuity, apply to it.
The matrix formula for the product of a matrix. In the coordinates relative to the
reference generators, the product of a and b is (a ⋆ b)ₖ = ∑ᵢ ∑ⱼ aᵢ bⱼ Mᵢⱼₖ.
The product of two reference basis vectors has the (i, j) entries of the matrix as its
coordinates: eᵢ ⋆ eⱼ = ∑ₖ Mᵢⱼₖ eₖ.
Every Novikov-bilinear map is the product of a matrix. A bilinear map over the Novikov ring between finitely generated completed modules is the product of the matrix of coordinates of its values on the reference basis vectors.