Normalized 2-cocycle for trivial group action

Definition

Suppose $G$ is a group and $A$ is an abelian group. A normalized 2-cocycle for trivial group action is a function $f : G \times G \to A$ that satisfies both these conditions:

$f$ is a 2-cocycle for trivial group action: $f (g, h k) + f (h, k) = f (g h, k) + f (g, h) \forall g, h, k, \in G$
The following equivalent conditions are satisfied:
- $f$ evaluates to the identity element of $A$ (written as 0 since $A$ is abelian) if either of its inputs is the identity element of $G$ : $f (g, 1) = f (1, g) = 0 \forall g \in G$
- $f$ evaluates to the identity element of $A$ (written as 0 since $A$ is abelian) if both of its inputs are the identity element of $G$ : $f (1, 1) = 0$

The set of normalized 2-cocycles for trivial group action form a subgroup (under pointwise addition) of the group of 2-cocycles for trivial group action. We will denote this subgroup as $Z_{n}^{2} (G, A)$ , though this is not standard notation. For the relationship with 2-coboundaries, constant functions, and second cohomology, see #Importance.

Important background fact that establishes equivalence of definitions

The equivalence of definitions follows from this background fact for any 2-cocycle $f : G \times G \to A$ , $f (x, 1) = f (1, x) = f (1, 1)$ for all $x \in G$ .

The fact that $f (x, 1) = f (1, 1)$ can be proved by setting $g = x, h = 1, k = 1$ in the identity $f (g, h k) + f (h, k) = f (g h, k) + f (g, h)$ and getting $f (x, 1) + f (1, 1) = f (x, 1) + f (x, 1)$ . One $f (x, 1)$ can be cancelled from both sides, leaving $f (1, 1) = f (x, 1)$ .
The fact that $f (1, 1) = f (1, x)$ can be proved by setting $g = 1, h = 1, k = x$ in the identity $f (g, h k) + f (h, k) = f (g h, k) + f (g, h)$ and getting $f (1, x) + f (1, x) = f (1, x) + f (1, 1)$ . One $f (1, x)$ can be cancelled from both sides, leaving $f (1, 1) = f (1, x)$ .

Importance

Normalizing a 2-cocycle

Translation normalization

The simplest normalization rule is a translation rule: given a 2-cocycle for trivial group action $f : G \times G \to A$ , we can obtain a normalized 2-cocycle $f_{0} : G \times G \to A$ defined as follows:

$f_{0} (g, h) = f (g, h) - f (1, 1)$

$f_{0}$ satisfies the identity for a 2-cocycle because it's just taking the original identity and subtracting $f (1, 1)$ twice from both sides.

$f_{0}$ is also normalized, because $f_{0} (g, 1) = f (g, 1) - f (1, 1)$ , which, based on the background fact, works out to zero.

Alternative normalization

Given a 2-cocycle for trivial group action $f : G \times G \to A$ , we can obtain a normalized 2-cocycle $f_{1} : G \times G \to A$ defined as follows:

$f_{1} (g, h) = f (g, h)$ if $g, h, g h$ are all non-identity elements
$f_{1} (g, 1) = f_{1} (1, g) = 0$ for all $g$
$f_{1} (g, g^{- 1}) = f (g, g^{- 1}) + f (1, 1)$ for all $g$ (note that it follows from the definition of 2-cocycle that $f (g, 1) = f (1, 1)$ for all $g$ , so this can also be written as $f_{1} (g, g^{- 1}) = f (g, g^{- 1}) + f (g, 1)$ )

$f_{1}$ is different from $f_{0}$ . As the later discussion will clarify, they differ by a 2-coboundary for trivial group action.

Quotient group of 2-cocycles by normalized 2-cocycles is the base group; in fact, a short exact sequence

There is a short exact sequence that splits canonically:

$0 \to Z_{n}^{2} (G, A) \to Z^{2} (G, A) \to A \to 0$

The injection from $Z_{n}^{2} (G, A)$ to $Z^{2} (G, A)$ is just treating a normalized 2-cocycle as a 2-cocycle. The surjection $Z^{2} (G, A) \to A$ maps $f$ to $f (1, 1)$ . This is a short exact sequence for the following reasons:

Injectivity of $Z_{n}^{2} (G, A) \to Z^{2} (G, A)$ : This is obvious
Composite $Z_{n}^{2} (G, A) \to Z^{2} (G, A) \to A$ is zero: This follows from the definition of normalized
Middle exactness: If $f (1, 1) = 0$ , then by the background fact above, $f (g, 1) = f (1, g) = 0$ as well, which is precisely the definition of normalized 2-cocycle.
Surjectivity and canonical splitting of $Z^{2} (G, A) \to A$ : For any element $a \in A$ , the constant function mapping everything to $A$ is a 2-cocycle (in fact, as we will see shortly, it is a 2-coboundary) so every element of $A$ is obtained this way.

Putting together 2-coboundaries, normalized 2-cocycles, and cocycles

Let's articulate the notation:

$Z^{2} (G, A)$ is a group of 2-cocycles for trivial group action
$Z_{n}^{2} (G, A)$ is the subgroup comprising normalized 2-cocycles
$B^{2} (G, A)$ is the subgroup comprising 2-coboundaries
$H^{2} (G, A)$ is the quotient $Z^{2} (G, A) / B^{2} (G, A)$ and is called the second cohomology group for trivial group action

We have two short exact sequences:

$0 \to Z_{n}^{2} (G, A) \to Z^{2} (G, A) \to A \to 0$

$0 \to B^{2} (G, A) \to Z^{2} (G, A) \to H^{2} (G, A) \to 0$

The first of these splits canonically as well! So it can be reversed, and we can write:

$0 \to A \to Z^{2} (G, A) \to Z_{n}^{2} (G, A) \to 0$

$0 \to B^{2} (G, A) \to Z^{2} (G, A) \to H^{2} (G, A) \to 0$

Imagine downward arrows from the terms of the first short exact sequence to the corresponding terms of the second one. The middle arrow is the identity map; the left arrow ( $A \to B^{2} (G, A)$ ) is injective, and the right arrow ( $Z_{n}^{2} (G, A) \to H^{2} (G, A)$ ) is surjective. Note that the lower short exact sequence does not have a canonical splitting.

Relation with treatment as central extensions

Let $E$ be a group with a central subgroup isomorphic to (and explicitly identified with) $A$ , and a quotient isomorphic to (and explicitly identified with) $G$ , such that the induced action of the quotient on the subgroup (in the sense of action by conjugation, see quotient group acts on abelian normal subgroup). Let $S$ be a system of coset representatives for $G$ in $E$ with $s : G \to S$ being the representation map. Then, define $f : G \times G \to A$ such that

$s (g h) = f (g, h) s (g) s (h)$

In other words, $f$ measures the extent to which the collection of coset representatives fails to be closed under multiplication.

Such an $f$ is a 2-cocycle for trivial group action of $G$ on $A$ . The different possible 2-cocycles for a given central extension form a single coset of the subgroup of 2-coboundaries, and hence a single element of the second cohomology group for trivial group action $H^{2} (G, A)$ .

If we further restrict $s$ so that $s (1)$ is the identity element of $A$ , i.e., the coset representative for the base subgroup is its identity element, then the 2-cocycle we obtain this way is a normalized 2-cocycle.

Since it's always possible to choose $s$ so that $s (1)$ is the identity element of $A$ , every central extension can be represented by a normalized 2-cocycle. In other words, every element of the second cohomology group for trivial group action $H^{2} (G, A)$ can be represented by a normalized 2-cocycle. Another way to frame this is that $Z^{2} (G, A)$ (the group of all 2-cocycles for trivial group action) is generated by $B^{2} (G, A)$ (the subgroup comprising 2-coboundaries) and the subgroup comprising normalized 2-cocycles.

We can explain the two normalizations discussed above as follows:

The translation normalization corresponds to taking the original system of coset representatives and multiplying all of them by $s (1)^{- 1}$ (left and right doesn't matter because $s (1)$ is central).
The alternative normalization corresponds to keeping all coset representatives intact except $s (1)$ , which is replaced by the identity element.