Continuous cohomology #
We define continuous cohomology as the homology of homogeneous cochains.
Implementation details #
We define homogenous cochains as g
-invariant continuous function in C(G, C(G,...,C(G, M)))
instead of the usual C(Gⁿ, M)
to allow more general topological groups other than locally compact
ones. For this to work, we also work in Action (TopModuleCat R) G
, where the G
action on M
is only continuous on M
, and not necessarily continuous in both variables, because the G
action
on C(G, M)
might not be continuous on both variables even if it is on M
.
For the differential map, instead of a finite sum we use the inductive definition
d₋₁ : M → C(G, M) := const : m ↦ g ↦ m
and
dₙ₊₁ : C(G, _) → C(G, C(G, _)) := const - C(G, dₙ) : f ↦ g ↦ f - dₙ (f (g))
See ContinuousCohomology.MultiInd.d
.
Main definition #
ContinuousCohomology.homogeneousCochains
: The functor taking anR
-linearG
-representation to the complex of homogeneous cochains.continuousCohomology
: The functor taking anR
-linearG
-representation to itsn
-th continuous homology.
TODO #
- Show that it coincides with
groupCohomology
for discrete groups. - Give the usual description of cochains in terms of
n
-ary functions for locally compact groups. - Show that short exact sequences induces long exact sequences in certain scenarios.
The G
representation C(G, rep)
given a representation rep
.
The G
action is defined by g • f := x ↦ g • f (g⁻¹ * x)
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The functor taking a representation rep
to the representation C(G, rep)
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The constant function rep ⟶ C(G, rep)
as a natural transformation.
Equations
- ContinuousCohomology.const R G = { app := fun (x : Action (TopModuleCat R) G) => { hom := TopModuleCat.ofHom (ContinuousLinearMap.const R G), comm := ⋯ }, naturality := ⋯ }
Instances For
The n-th functor taking M
to C(G, C(G,...,C(G, M)))
(with n G
s).
These functors form a complex, see MultiInd.complex
.
Equations
Instances For
The differential map in MultiInd.complex
.
Equations
- One or more equations did not get rendered due to their size.
- ContinuousCohomology.MultiInd.d R G 0 = ContinuousCohomology.const R G
Instances For
The complex of functors whose behaviour pointwise takes an R
-linear G
-representation M
to the complex M → C(G, M) → ⋯ → C(G, C(G,...,C(G, M))) → ⋯
The G
-invariant submodules of it is the homogeneous cochains (shifted by one).
Equations
Instances For
The functor taking an R
-linear G
-representation to its G
-invariant submodule.
Equations
- One or more equations did not get rendered due to their size.
Instances For
homogeneousCochains R G
is the functor taking
an R
-linear G
-representation to the complex of homogeneous cochains.
Equations
- One or more equations did not get rendered due to their size.
Instances For
continuousCohomology R G n
is the functor taking
an R
-linear G
-representation to its n
-th continuous homology.
Equations
Instances For
The 0
-homogeneous cochains are isomorphic to Xᴳ
.
Equations
- One or more equations did not get rendered due to their size.
Instances For
H⁰_cont(G, X) ≅ Xᴳ
.
Equations
- One or more equations did not get rendered due to their size.