Transfer in group homology #
Let S be a finite-index subgroup of a group G. Group homology has a transfer map
H_n(G, M) ⟶ H_n(S, Resˢᴳ M).
Unlike the covariant map induced by the inclusion S → G, the transfer goes against the group
homomorphism. It is obtained from the unit M ⟶ Indˢᴳ Resˢᴳ M of the finite-index
adjunction, followed by Shapiro's isomorphism
H_n(G, Indˢᴳ Resˢᴳ M) ≃ H_n(S, Resˢᴳ M). Transporting it across the negative-degree
comparison gives restriction in Tate cohomology below degree -1.
Followed by corestriction H_n(S, Resˢᴳ M) ⟶ H_n(G, M), the map induced by the inclusion, the
transfer is multiplication by the index [G : S]. Read through Shapiro's isomorphism,
corestriction is the map induced by the counit Indˢᴳ Resˢᴳ M ⟶ M
(TauCeti.groupHomology.indIso_inv_comp_map_counit), and the unit followed by the counit is
[G : S].
Taking for S the trivial subgroup of a finite group G, whose positive-degree homology
vanishes, shows that positive-degree homology of a finite group is killed by the order of the
group, and vanishes when multiplication by that order is bijective on the coefficients. With
trivial integral coefficients, this says that the integral homology of a finite group is torsion
in positive degrees.
Main definitions #
TauCeti.groupHomology.transfer: the transfer from a group to a finite-index subgroup.
Main results #
TauCeti.groupHomology.transfer_comp_indIso_inv: through the inverse of Shapiro's isomorphism, transfer is the map induced by the unit of the finite-index adjunction.TauCeti.groupHomology.map_comp_transfer: the transfer is natural in the coefficients.TauCeti.groupHomology.transfer_comp_map_subtype_id: corestriction after transfer is multiplication by the index[G : S].TauCeti.groupHomology.index_nsmul_eq_zero_of_transfer_eq_zero: a class whose transfer toSvanishes is killed by[G : S].groupHomology.natCard_nsmul_eq_zero: positive-degree homology of a finite group is killed by the order of the group.groupHomology.subsingleton_of_bijective_nsmul,groupHomology.subsingleton_of_isUnit_natCard: positive-degree homology of a finite group vanishes when multiplication by the order of the group is bijective on the coefficients, in particular when the order is a unit of the coefficient ring.TauCeti.groupHomology.transfer_zero_H0π: in degree zero, where group homology is the module of coinvariants, the transfer is the relative transferRepresentation.relTransfer,⟦m⟧ ↦ ⟦∑_{q ∈ G ⧸ S} q⁻¹ • m⟧.
References #
- K. S. Brown, Cohomology of Groups, Chapter III, Sections 9–10.
- E. Artin and J. Tate, Class Field Theory, Chapter IV, §6.
The transfer in group homology from a group to a finite-index subgroup. It is the map induced
by the unit M ⟶ Indˢᴳ Resˢᴳ M, followed by the homological Shapiro isomorphism.
Instances For
Through the inverse of the homological Shapiro isomorphism, transfer is the map induced by the unit of the finite-index induction--restriction adjunction.
Through the inverse of the homological Shapiro isomorphism, transfer is the map induced by the unit of the finite-index induction--restriction adjunction.
The transfer is natural in the coefficients: for a morphism φ : M ⟶ N of
G-representations, transfer intertwines the map Hₙ(G, M) ⟶ Hₙ(G, N) induced by φ with the
map Hₙ(S, Res_S M) ⟶ Hₙ(S, Res_S N) induced by its restriction to S. The cohomological
counterpart, for corestriction, is TauCeti.groupCohomology.map_comp_corestriction.
The transfer is natural in the coefficients: for a morphism φ : M ⟶ N of
G-representations, transfer intertwines the map Hₙ(G, M) ⟶ Hₙ(G, N) induced by φ with the
map Hₙ(S, Res_S M) ⟶ Hₙ(S, Res_S N) induced by its restriction to S. The cohomological
counterpart, for corestriction, is TauCeti.groupCohomology.map_comp_corestriction.
The transfer is natural in the coefficients: for a morphism φ : M ⟶ N of
G-representations, transfer intertwines the map Hₙ(G, M) ⟶ Hₙ(G, N) induced by φ with the
map Hₙ(S, Res_S M) ⟶ Hₙ(S, Res_S N) induced by its restriction to S. The cohomological
counterpart, for corestriction, is TauCeti.groupCohomology.map_comp_corestriction.
Corestriction after transfer is multiplication by the index: for a finite-index subgroup
S ≤ G, the composite Hₙ(G, M) ⟶ Hₙ(S, Res_S M) ⟶ Hₙ(G, M) of the transfer and corestriction
is [G : S] times the identity, in every degree n.
Corestriction after transfer is multiplication by the index: for a finite-index subgroup
S ≤ G, the composite Hₙ(G, M) ⟶ Hₙ(S, Res_S M) ⟶ Hₙ(G, M) of the transfer and corestriction
is [G : S] times the identity, in every degree n.
Corestriction after transfer is multiplication by the index: for a finite-index subgroup
S ≤ G, the composite Hₙ(G, M) ⟶ Hₙ(S, Res_S M) ⟶ Hₙ(G, M) of the transfer and corestriction
is [G : S] times the identity, in every degree n.
A class in Hₙ(G, M) whose transfer to the finite-index subgroup S vanishes is killed by
the index of S.
In degree zero, the transfer is the relative transfer on coinvariants. Group homology in
degree zero is the module of coinvariants, and the transfer H₀(G, M) ⟶ H₀(S, Res_S M) sends the
class of m to the class of ∑_{q ∈ G ⧸ S} q⁻¹ • m, the relative transfer
Representation.relTransfer.
Positive-degree homology of a finite group is killed by the order of the group: for every
representation A of a finite group G, #G • x = 0 for every x ∈ Hₙ₊₁(G, A). The transfer to
the trivial subgroup lands in the vanishing homology of the trivial group.
Positive-degree homology of a finite group vanishes when multiplication by the order of the
group is bijective on the coefficients, for instance for a representation on a ℚ-vector space
over any coefficient ring.
Positive-degree homology of a finite group vanishes when the order of the group is a unit of the coefficient ring.