Associativity of the cup product on continuous cohomology #
Let μ₁ : TopPairing A B D, μ₂ : TopPairing D C E, ν₁ : TopPairing B C F and
ν₂ : TopPairing A F E be four coefficient pairings of topological representations of a
topological group G, compatible in the sense that
μ₂ (μ₁ a b) c = ν₂ a (ν₁ b c) for all a : A, b : B, c : C.
Then the two parenthesizations of a threefold cup product agree on continuous cohomology:
(x ⌣_{μ₁} y) ⌣_{μ₂} z = x ⌣_{ν₂} (y ⌣_{ν₁} z)
for x ∈ Hᵐ(G, A), y ∈ Hⁿ(G, B) and z ∈ Hᵖ(G, C), where the right-hand side, which lives in
degree m + (n + p), is transported to the degree m + n + p of the left-hand side
(TauCeti.TopPairing.cup_assoc). Four pairings are needed to type the two sides at all, and the
single-ring specialization that the applications use, one coefficient object X with an
associative pairing P : TopPairing X X X in all four places, is the instance
cup_assoc P P P P. For a discrete G-ring S acted on by ring automorphisms, with all four
pairings its multiplication TauCeti.ofDiscreteModulePairing AddMonoidHom.mul, this instance is
stated separately as TauCeti.TopPairing.cup_assoc_mul: associativity of the cohomology ring
H^•(G, S).
The identity already holds on homogeneous cochains, before any passage to cohomology. The Alexander–Whitney formula
(a ⌣ b) (g₀, …, g_{m+n}) = μ (a (g₀, …, g_m)) (b (g_m, …, g_{m+n}))
pairs the leading m + 1 arguments into a and the trailing n + 1 into b, so both
parenthesizations of a ⌣ b ⌣ c evaluate to
μ₂ (μ₁ (a (g₀, …, g_m)) (b (g_m, …, g_{m+n}))) (c (g_{m+n}, …, g_{m+n+p}))
and the two sides differ only by the coefficient identity. This is Brown's cochain-level
associativity (V (3.5)); no chain homotopy is involved, in contrast with graded commutativity,
which fails on cochains. On the curried coinduced resolution the proof is an induction along the
recursion (a ⌣ b) g = (a g) ⌣ b of TauCeti.TopPairing.resolutionCup, whose base cases are the
corresponding identities for the pointwise pairing TauCeti.TopPairing.pointwise.
Degrees #
The resolution-level identity TauCeti.TopPairing.resolutionCup_assoc is stated, as everything in
TauCeti.RepresentationTheory.Homological.ContCohomology.Cup.Graded.Basic, with the total degrees
as explicit arguments together with the equations they satisfy, so that both parenthesizations
land in the same term of the resolution and no transport appears. The bilinear forms
TauCeti.TopPairing.resolutionCupPairing_assoc and TauCeti.TopPairing.cupCochain_assoc land in
degrees m + n + p and m + (n + p), which are equal but not definitionally, so the right-hand
side is transported through HomologicalComplex.XIsoOfEq; on cohomology the transport is
TauCeti.ContinuousCohomology.degreeCast.
Main results #
TauCeti.TopPairing.resolutionCup_assoc: associativity of the Alexander–Whitney pairing on the coinduced resolution.TauCeti.TopPairing.cupCochain_assoc: associativity of the cup product of homogeneous cochains, an identity of cochains and not only of classes.TauCeti.TopPairing.cup_assoc: associativity of the cup product on continuous cohomology.TauCeti.TopPairing.cup_assoc_mul: the specialization to a discreteG-ring with its multiplication as the pairing, associativity of the cohomology ring.
References #
- K. S. Brown, Cohomology of Groups, GTM 87, Springer (1982), Chapter V, §3, (3.5), for associativity at the cochain level.
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Springer (2008), Chapter I, §4, (1.4.4).
Associativity on the resolution #
Associativity of the Alexander–Whitney pairing on the coinduced resolution, with explicit
total degrees: (a ⌣_{μ₁} b) ⌣_{μ₂} c = a ⌣_{ν₂} (b ⌣_{ν₁} c) for compatible pairings. Both sides
live in the (k + 1)-st term of the resolution of E, for k = p + (n + m) = (p + n) + m.
Associativity of the Alexander–Whitney pairing, as bilinear maps: the right-hand side lives
in degree m + (n + p) and is transported to m + n + p.
Associativity on cochains and on cohomology #
Associativity of the cup product of homogeneous cochains:
(a ⌣_{μ₁} b) ⌣_{μ₂} c = a ⌣_{ν₂} (b ⌣_{ν₁} c) for compatible pairings, as an identity of
cochains, where the right-hand side lives in degree m + (n + p) and is transported to
m + n + p.
Associativity of the cup product on continuous cohomology: for compatible pairings
μ₂ (μ₁ a b) c = ν₂ a (ν₁ b c) and classes x ∈ Hᵐ(G, A), y ∈ Hⁿ(G, B), z ∈ Hᵖ(G, C),
(x ⌣_{μ₁} y) ⌣_{μ₂} z = x ⌣_{ν₂} (y ⌣_{ν₁} z),
where the right-hand side lives in degree m + (n + p) and is transported to m + n + p. For one
coefficient object X with an associative pairing P : TopPairing X X X in all four places, this
is associativity of the cup product of the cohomology ring H^•(G, X).
Associativity for a discrete G-ring #
The specialization the applications use: one discrete coefficient ring S acted on by ring
automorphisms, all four pairings its multiplication AddMonoidHom.mul, and the coefficient
identity mul_assoc. The equivariance a discrete-module pairing has to come with is smul_mul',
which applies to AddMonoidHom.mul as it stands, exactly as for the low-degree
TauCeti.ContCohomology.explicitCup_assoc000_mul and its siblings.
Associativity of the cup product of the cohomology ring of a discrete G-ring: for a
discrete ring S acted on by ring automorphisms and classes x ∈ Hᵐ(G, S), y ∈ Hⁿ(G, S),
z ∈ Hᵖ(G, S), with the multiplication of S as the coefficient pairing throughout,
(x ⌣ y) ⌣ z = x ⌣ (y ⌣ z), where the right-hand side lives in degree m + (n + p) and is
transported to m + n + p. This is TauCeti.TopPairing.cup_assoc at the pairing
TauCeti.ofDiscreteModulePairing AddMonoidHom.mul with coefficient identity mul_assoc.