Group cohomology ( $H^{n} (G, M)$ )

Group cohomology studies a group through cochain complexes built from its actions on abelian groups. In degree one,

H^{1} (G, M)

identifies one-cocycles

c : G \to M

modulo one-coboundaries.

Table of contents

Projective-resolution definition of group cohomology

Choose a projective resolution

P_{∙} \to Z

of the trivial

Z G

-module. For a

Z G

-module

M

, group cohomology is

H^{n} (G, M) = H^{n} (Hom_{Z G} (P_{∙}, M)) = Ext_{Z G}^{n} (Z, M) .

Different projective resolutions give naturally isomorphic groups.

Group cohomology commutes with finite direct sums

For

Z G

-modules

M_{1}, M_{2}

,

H^{n} (G, M_{1} \oplus M_{2}) ≅ H^{n} (G, M_{1}) \oplus H^{n} (G, M_{2}),

because the Hom functor into a finite direct sum splits degree by degree and taking cohomology preserves that splitting.

Coinduced module ( $Coind_{K}^{G} X$ )

K \leq G

and a

Z K

-module

X

, the coinduced module is

Coind_{K}^{G} X = Hom_{Z K} (Z G, X), (g f) (r) = f (r g) .

Shapiro's lemma

Shapiro's lemma gives a natural isomorphism

H^{n} (G, Coind_{K}^{G} X) ≅ H^{n} (K, X) .

It follows by applying the Hom functor adjunction for a coinduced module to a projective resolution and observing that restriction from

G

to

K

preserves projective modules.

Hom functor adjunction for a coinduced module

For a

Z G

-module

P

and a

Z K

-module

X

,

Hom_{Z G} (P, Hom_{Z K} (Z G, X)) ≅ Hom_{Z K} (P, X) .

1 \in G

gives the forward map. If

a : P \to X

is

Z K

-linear, its inverse sends

a

to

p \mapsto (r \mapsto a (r p))

.

Conjugation module of a group ring

Z G

becomes a

Z G

-module under the conjugation action

g \cdot x = gx g^{- 1}

. Its basis splits into conjugacy classes, and the span of the class of

x

is the permutation module on

G / C_{G} (x)

.

Group cohomology of a conjugation module

For a finite group

G

and representatives

g_{i}

of its conjugacy classes, Shapiro's lemma gives

H^{n} (G, Z G_{conj}) ≅ ⨁_{i} H^{n} (C_{G} (g_{i}), Z) .

Integral cohomology of a finite cyclic group

For the trivial action of the finite cyclic group

C_{m}

on

Z

,

H^{0} (C_{m}, Z) = Z, H^{2 k + 1} (C_{m}, Z) = 0, H^{2 k} (C_{m}, Z) = Z / m Z

for

k \geq 1

. The periodic resolution of a finite cyclic group proves this directly.

Periodic resolution of a finite cyclic group

If

C_{m} = ⟨ t ⟩

and

N = 1 + t + \dots + t^{m - 1}

, a free resolution of the trivial

Z C_{m}

-module alternates multiplication by

t - 1

and

N

. Applying

Hom_{Z C_{m}} (-, Z)

for the trivial action alternates the zero map and multiplication by

m

.

Group extension

A group extension of

G

by an abelian

G

-module

A

is a short exact sequence

1 ⟶ A ⟶ E ⟶ G ⟶ 1

whose conjugation action on

A

is the specified

G

Second group cohomology classifies group extensions

Equivalence classes of group extensions of

G

by an abelian

G

-module

A

correspond to

H^{2} (G, A)

. A section

s : G \to E

produces the extension cocycle

c (g, h) = s (g) s (h) s (g h)^{- 1},

and changing the section changes

c

by a group coboundary.

Extension cocycle

An extension cocycle is the two-cocycle obtained from a section of a group extension. It measures the failure of the section to be a group homomorphism.

Group cocycle

A group cocycle is an element of the kernel of the coboundary map in the standard cochain complex computing group cohomology.

One-cocycle

For a

G

-module

M

, a one-cocycle is a map

f : G \to M

satisfying

f (g h) = f (g) + g \cdot f (h)

.

Two-cocycle

For a

G

-module

M

, a two-cocycle is a map

c : G \times G \to M

satisfying

g \cdot c (h, k) - c (g h, k) + c (g, hk) - c (g, h) = 0.

Group coboundary

A group coboundary is a cochain in the image of the preceding coboundary map. Quotienting group cocycles by group coboundaries gives group cohomology.

Lyndon–Hochschild–Serre spectral sequence

For a normal subgroup

H ◃ G

Q = G / H

, and a

G

-module

M

, the Lyndon–Hochschild–Serre spectral sequence has

E_{2}^{p, q} = H^{p} (Q, H^{q} (H, M)) ⟹ H^{p + q} (G, M) .

Five-term exact sequence in group cohomology

The low-degree edge maps of the Lyndon–Hochschild–Serre spectral sequence give

0 \to H^{1} (Q, M^{H}) inf H^{1} (G, M) res H^{1} (H, M)^{Q} d_{2} H^{2} (Q, M^{H}) inf H^{2} (G, M) .

Inflation map in group cohomology

Inflation composes a cocycle on the quotient

Q = G / H

with the quotient map

G \to Q

and regards its values in the invariant submodule

M^{H}

as values in

M

.

Restriction map in group cohomology

Restriction sends a cocycle on

G

to its restriction to a subgroup

H

. If

H

is normal, its image in

H^{1} (H, M)

is fixed by the induced

G / H

Transgression in group cohomology

The transgression

d_{2} : H^{1} (H, M)^{Q} \to H^{2} (Q, M^{H})

is the first differential crossing from the vertical to the horizontal edge of the Lyndon–Hochschild–Serre spectral sequence. If a representative on

H

is extended to a cochain on

G

, its coboundary descends to the representing two-cocycle on

Q

.

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Group cohomology by

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Group cohomology is a mathematical tool used in algebraic topology, group theory, and various other areas of mathematics. It provides a way to study the properties of groups using cohomological methods, which are analogous to those used in homology theory but focus on the algebraic structure associated with groups.

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