Linear maps of completed combinations given by supported kernels #
Let F₁ : S₁ → ℝ and F₀ : S₀ → ℝ be potentials, and let m : S₁ → S₀ → R be a kernel:
m p₁ p₀ is the coefficient of the output generator p₀ in the image of the input generator
p₁. Suppose that
m p₁ p₀ ≠ 0impliesF₀ p₀ ≤ F₁ p₁ + Cfor a fixed constantC, and- each output function
m p₁is itself a completed combination.
Then for a completed combination a the sum ∑ a(p₁) m(p₁, p₀) has finitely many nonzero terms,
and it defines a linear map NovikovModule R F₁ → NovikovModule R F₀
(NovikovModule.kernelMap), which shifts potential cutoffs by C.
Floer and Novikov differentials, continuation maps, chain homotopies and comparison maps are all
of this form: the generators are lifted critical points or capped orbits, the potentials are the
action functionals, m counts rigid trajectories, and the potential bound is the energy estimate.
The kernel of a composite is the matrix product ∑ m(p₁, p₀) m'(p₀, p₂) of the two kernels
(NovikovModule.kernelComp). It is again supported, with the sum of the two constants, and the
composite of the two linear maps is the linear map of the composite kernel
(NovikovModule.kernelMap_kernelMap). Hence identities between chain maps, such as d ∘ d = 0
or f ∘ g - id = d ∘ h + h ∘ d, follow from the corresponding identities between kernels.
Suppose next that each Sₗ is a finite disjoint union of torsors under a group H with
equivariant potentials, and that the kernel is unchanged when the input and the output are
translated by the same element. In lift coordinates the map is given by the matrix of Novikov
series of the kernel in the reference generators (NovikovModule.coordinates_kernelMap), so it is
linear over the Novikov ring (NovikovModule.novikovKernelMap). No discreteness or injectivity of
the periods is assumed.
Main definitions #
TauCeti.NovikovModule.kernelMap: the linear map given by a supported kernel.TauCeti.NovikovModule.kernelComp: the composite of two kernels.TauCeti.NovikovModule.toKernel: the matrix coefficients of a linear map.TauCeti.NovikovModule.novikovKernelMap: the map of an invariant kernel, as a linear map over the Novikov ring.
Main results #
TauCeti.NovikovModule.kernelMap_apply_val_eq_zero: the map shifts potential cutoffs by the constant of the potential bound.TauCeti.NovikovModule.potential_le_of_kernelComp_ne_zero,TauCeti.NovikovModule.kernelComp_mem: the composite kernel is supported.TauCeti.NovikovModule.kernelMap_kernelMap: composition of maps is composition of kernels.TauCeti.NovikovModule.kernelMap_add,TauCeti.NovikovModule.kernelMap_zero: the map is additive in the kernel.TauCeti.NovikovModule.kernelMap_sub_kernelMap: a kernel homotopy between two kernels gives a chain homotopy between their maps.TauCeti.NovikovModule.kernelMap_eq_id: the identity kernel gives the identity map, as the special case of a diagonal kernel (TauCeti.NovikovModule.kernelMap_diagonal_apply_val).TauCeti.NovikovModule.kernelMap_inj: a kernel map determines its kernel, through the images of point masses (TauCeti.NovikovModule.kernelMap_single_apply_val).TauCeti.NovikovModule.coordinates_kernelMap: the matrix formula in lift coordinates.
References #
- M. Hutchings and Y.-J. Lee, Circle-valued Morse theory and Reidemeister torsion, Geom. Topol. 3 (1999), Section 2.
- A. Pajitnov, Circle-valued Morse Theory, de Gruyter 2006, Chapter 3.
- D. Salamon, Lectures on Floer homology, in Symplectic geometry and topology, IAS/Park City Math. Ser. 7 (1999), 143–229.
Under a potential bound, only finitely many terms of a sum of kernel values weighted by a completed combination are nonzero.
Summing completed outputs of a kernel with a potential bound against a completed combination gives a completed combination.
For a kernel obeying a potential bound, each coefficient of the image is a sum with finitely many nonzero terms.
The linear map of completed combinations given by a kernel m with potential bound C and
completed outputs: the coefficient of p₀ in the image of a is the finite sum
∑ a(p₁) m(p₁, p₀) over all input generators.
Instances For
The coefficient of a generator in an image is the sum, over all input generators, of the product of their coefficients with the kernel.
A coefficient of an image is a sum over any finite set containing every support generator with nonzero kernel coefficient.
A kernel map sends a generator to its row of the kernel.
A kernel whose rows have finitely many nonzero entries maps finite combinations to finite combinations.
The map shifts potential cutoffs by the constant of the potential bound: if a vanishes
above D, then its image vanishes above D + C.
The map of a sum of kernels is the sum of their maps. The three potential bounds may have different constants.
A diagonal kernel, with coefficient s p from p to itself and 0 off the diagonal,
satisfies the potential bound with constant 0.
A diagonal kernel has completed outputs.
A diagonal kernel, with coefficient s p from p to itself and 0 off the diagonal,
multiplies each coefficient by s.
The identity kernel, with coefficient 1 on the diagonal and 0 off it, gives the identity
map.
The image of the point mass r at an input generator p₁ has the coefficients r * m p₁
of the output of the kernel at p₁.
A kernel map determines its kernel. Two kernel maps, possibly with different potential bounds, coincide exactly when their kernels do.
The kernel of a linear map of completed combinations: the coefficient of the output
generator p₀ in the image of the input generator p₁.
Instances For
Each row of the kernel of a linear map is a completed combination.
The kernel of a kernel map is its kernel.
The composite of a kernel m from S₁ to S₀ with a kernel m' from S₀ to S₂: the
matrix product ∑ m(p₁, p₀) m'(p₀, p₂) over the intermediate generators.
Instances For
Potential bound of a composite kernel. If the outputs of m at p₁ have potential at
most D and m' raises potentials by at most C', then the outputs of the composite kernel at
p₁ have potential at most D + C'.
Completed outputs of a composite kernel. If the output of m at p₁ is a completed
combination and m' is supported, then the output of the composite kernel at p₁ is a completed
combination.
Composition of kernel maps. Applying the maps of m and then of m' is the map of the
composite kernel. The potential bound of the composite may have any constant.
Composition of kernel maps, as an equation of linear maps.
Chain homotopies from kernel homotopies. Let δ₁ and δ₀ be kernels on the input and
the output generators. If a kernel k is a kernel homotopy between kernels m and m',
m - m' = k δ₀ + δ₁ k as composite kernels, then its kernel map K is a homotopy between their
maps: m a - m' a = δ₀ (K a) + K (δ₁ a) for every completed combination a. Continuation maps
compared by a one-parameter family of Floer data are related in this way.
In lift coordinates, a completed combination given by an invariant one-input kernel m,
c(p₀) = ∑ a(p₁) m(p₁, p₀), is given by the matrix of Novikov series of m in the reference
generators: cₖ = ∑ᵢ aᵢ Mᵢₖ.
The matrix of an invariant kernel map. In lift coordinates, the map of an invariant
kernel is multiplication by the matrix whose (i, k) entry is the Novikov series of coordinates,
in the orbit k, of the output of m at the reference generator of the orbit i.
The map of an invariant kernel is linear over the Novikov ring, for the Novikov module
structures of TauCeti.NovikovModule.module.
The map of an invariant kernel, as a linear map over the Novikov ring, for the Novikov module
structures of TauCeti.NovikovModule.module.
Instances For
The Novikov-linear map evaluates to the map of the kernel.