Coboundary shifts of the coinduced resolution by chain endomorphisms #
Let X be a smooth discrete representation of a compact group G, and let z and w be
homogeneous elements of the total coinduced resolution X.gradedResolutionTopRep, of degrees
m + 1 and m, both fixed by G. If linear maps from the total resolution to its degree m
term take every value at z, which holds for every z ≠ 0 over a field and also, when X is
killed by 2, over any commutative ring in which 2 generates a maximal ideal, such as ℤ, then
there is an equivariant endomorphism Φ of the total resolution, of degree zero, commuting with
the differential d and fixing the augmentation, with
Φ z = z + d w.
So over a field, or with 𝔽₂ coefficients over ℤ, a nonzero invariant cocycle can be moved, by
a chain endomorphism over the identity of X, to any cocycle differing from it by the differential
of an invariant element. This is useful for constructions on representing cocycles that are not
additive, such as Evens' multiplicative transfer, the class of the tensor power z ⊗ ⋯ ⊗ z: a
construction that is functorial in chain maps over the identity and whose classes are invariant
under them (TopRep.ResolutionContraction.gradedCohomologyClass_map) then depends on z only
through its class.
The endomorphism is Φ = 1 + d ∘ h + h ∘ d for an equivariant h of degree -1 with h z = w,
so Φ is homotopic to the identity; what matters is the exact equation Φ z = z + d w, not only
the equality of classes. Such an h exists because the degree-m term C(G, Y) of the resolution
is coinduced: by Frobenius reciprocity (ContRepresentation.coind₁Lift), a linear map φ into
Y with φ z = w 1 lifts to the equivariant map v ↦ (x ↦ x • φ (x⁻¹ • v)) into C(G, Y),
which sends the invariant vector z to the invariant element x ↦ x • w 1 = w x.
Main results #
TopRep.exists_gradedResolutionTopRep_hom_apply_eq_add_of_forall_exists_apply_eq: a chain endomorphism over the identity withΦ z = z + d w, forzat which linear maps to the degreemterm take every value.TopRep.exists_gradedResolutionTopRep_hom_apply_eq_add: the same over a field, for everyz ≠ 0.TopRep.exists_gradedResolutionTopRep_hom_apply_eq_add_of_add_self_eq_zero: the same for everyz ≠ 0whenXis killed by2and2generates a maximal ideal of the coefficient ring.
References #
- L. Evens, A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), §§2–5, for the multiplicative transfer computed on representing cocycles.
Realizing a coboundary shift by a chain endomorphism. Let z and w be homogeneous
elements of degrees m + 1 and m of the total coinduced resolution of a smooth discrete
representation X of a compact group, both fixed by G, and suppose that linear maps from the
total resolution to its degree m term take every value at z; a linear functional f with
f z = 1 gives v ↦ f v • c with value c. Then some endomorphism Φ of the total resolution
commutes with the differential, fixes the augmentation, has degree zero, and satisfies
Φ z = z + d w.
Realizing a coboundary shift by a chain endomorphism, over a field. Let z ≠ 0 and w be
homogeneous elements of degrees m + 1 and m of the total coinduced resolution of a smooth
discrete representation X of a compact group, both fixed by G. Then some endomorphism Φ of
the total resolution commutes with the differential, fixes the augmentation, has degree zero, and
satisfies Φ z = z + d w.
Realizing a coboundary shift by a chain endomorphism, in characteristic two. Let R be a
commutative ring in which 2 generates a maximal ideal, such as ℤ or a field of characteristic
two, and let X be a smooth discrete representation of a compact group over R whose elements are
their own additive inverses, such as trivial 𝔽₂ coefficients over ℤ. Let z ≠ 0 and w be
homogeneous elements of degrees m + 1 and m of its total coinduced resolution, both fixed by
G. Then some endomorphism Φ of the total resolution commutes with the differential, fixes the
augmentation, has degree zero, and satisfies Φ z = z + d w.