Normalized 2-cocycle for trivial group action

Definition

Equivalent definitions in tabular format

Suppose $G$ is a group and $A$ is an abelian group. We denote by $1$ the identity element of $G$ and by $0$ the identity element of $A$; the group operation for $G$ is denoted multiplicatively and the group operation for $A$ is denoted additively. A normalized 2-cocycle for trivial group action is a function $f:G\times G\to A$ that satisfies the following equivalent conditions:

No.	Shorthand	Full statement
1	2-cocycle for trivial group action that is zero along the axes	$f$ is a 2-cocycle for trivial group action: $f(g,hk)+f(h,k)=f(gh,k)+f(g,h)\mathrm{\forall }g,h,k,\in G$ and $f(g,1)=f(1,g)=0\mathrm{\forall }g\in G$.
2	2-cocycle for trivial group action that is zero at origin	$f$ is a 2-cocycle for trivial group action: $f(g,hk)+f(h,k)=f(gh,k)+f(g,h)\mathrm{\forall }g,h,k,\in G$ and $f(1,1)=0$.
3	System of coset representatives where identity is represented by identity	$f$ corresponds to a system of coset representatives for a central extension $E$ (corresponding to an element of $H^2(G,A)$, the second cohomology group for trivial group action) where the coset representative of the identity element is the identity element.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

The set of normalized 2-cocycles for trivial group action forms 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.

Equivalence of definitions

The equivalence of definitions (1) and (2) follows from 2-cocycle for trivial group action is constant on axes, which states that $f(g,1)=f(1,g)=f(1,1)\mathrm{\forall }g\in G$.

The equivalence with definition (3) is discussed in more detail at #Relation with central extensions.

Importance

Normalizing a 2-cocycle (finding a normalized 2-cocycle that differs from it via a 2-coboundary)

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 $f_0(g,1)=f(g,1)-f(1,1)$ and $f_0(1,g)=f(1,g)-f(1,1)$, which are both zero based on 2-cocycle for trivial group action is constant on axes.

Moreover, the difference of $f$ and $f_0$ is a constant function $G\times G\to A$ with constant value $f(1,1)$, which is therefore also a 2-coboundary (of the constant function $G\to A$ taking value $f(1,1)$. Thus, $f_0$ belongs to the same cohomology class as $f$, i.e., they project to the same element of the second cohomology group for trivial group action $H^2(G,A)$.

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 $g,h,gh$ 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.

For most real-world purposes, we go with $f_0$. However, $f_1$ could be useful if we want to preserve as many of the values of $f$ as possible.

The group of 2-cocycles is a direct sum of the subgroup of normalized 2-cocycles and the base group (realized as constant functions)

We have:

$Z^2(G,A)=Z_n^2(G,A)\oplus A$

This is a canonical split as an internal direct sum. The terms are as follows:

• $Z^2(G,A)$ is the group of all 2-cocycles for trivial group action.
• $Z_n^2(G,A)$ is the subgroup comprising normalized 2-cocycles for trivial group action.
• $A$ is identified with the subgroup comprising constant functions $G\times G\to A$. Specifically, an element $a\in A$ corresponds to the constant function $c_a$ sending everything in $G\times G$ to $a$.

To show that this is an internal direct sum, we need to verify the following:

• $Z_n^2(G,A)$ is a subgroup of $Z^2(G,A)$: This is obvious from the definition.
• $A$ (identified as constant functions) is a subgroup of $Z^2(G,A)$: This is also obvious from the definition of 2-cocycle.
• $Z_n^2(G,A)\cap A=0$: For $f$ a normalized 2-cocycle, $f(1,1)=0$. If $f$ is also constant, this forces it to be zero everywhere.
• $Z_n^2(G,A)+A=Z^2(G,A)$, i.e., every 2-cocycle can be expressed as the sum of a normalized 2-cocycle and a constant function: We already demonstrated this in the preceding section in the form of translation normalization.

Some implications of this:

• $Z^2(G,A)/Z_n^2(G,A)\cong A$
• $Z^2(G,A)/A\cong Z_n^2(G,A)$ (where the group $A$ in the denominator is realized as constant functions)
• $0\to Z_n^2(G,A)\to Z^2(G,A)\to A\to 0$ is a canonically split short exact sequence.
• $0\to A\to Z^2(G,A)\to Z_n^2(G,A)\to 0$ is a canonically split short exact sequence.

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.
• $B_n^2(G,A)$ is the subgroup comprising normalized 2-coboundaries; it is the intersection $Z_n^2(G,A)\cap B^2(G,A)$.
• $A$ is identified with constant functions $G\times G\to A$.
• $H^2(G,A)$ 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, the first of which splits canonically and was discussed in the preceding section:

$0\to A\to Z^2(G,A)\to Z_n^2(G,A)\to 0$

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

The new fact of interest is that all constant functions are also 2-coboundaries! That's because the constant function $G\times G\to A$ taking value $a\in A$ can be realized as the coboundary of the constant function $G\to A$ taking value $a$. This means that $A$ maps injectively into $B^2(G,A)$. This allows us to make a downward arrow from $A$ to $B^2(G,A)$ that is injective. We can also make a downward arrow between the middle groups $Z^2(G,A)$ that is the identity map. This in turn allows us to make a downward map from $Z_n^2(G,A)$ to $H^2(G,A)$, and shows that this map must be surjective. In other words, every cohomology class can be represented by a normalized 2-cocycle. The kernel of this map is $B_n^2(G,A)$, so we obtain this short exact sequence:

$0\to B_n^2(G,A)\to Z_n^2(G,A)\to H^2(G,A)\to 0$

We also have a canonical internal direct sum splitting:

$B^2(G,A)=B_n^2(G,A)\oplus A$

This uses the same logic as the splitting for $Z^2(G,A)$ from the preceding section, and the fact that all the constant functions are in fact 2-coboundaries.

We can put all this information together in the commutative diagram below, whose rows are short exact sequences and the last column is also a short exact sequence; the first two rows also have canonical splittings:

$\left(\begin{array}{cccccccc}& & & & & 0& & \\ & & & & & \downarrow & & \\ 0\to & A& \to & Z_n^2(G,A)& \to & B_n^2(G,A)& \to & 0\\ & \downarrow & & \downarrow & & \downarrow & & \\ 0\to & A& \to & Z^2(G,A)& \to & Z_n^2(G,A)& \to & 0\\ & \downarrow & & \downarrow & & \downarrow & & \\ 0\to & B^2(G,A)& \to & Z^2(G,A)& \to & H^2(G,A)& \to & 0\\ & & & & & \downarrow & & \\ & & & & & 0& & \\ \end{array}\right)$

Relation with treatment as central extensions

This subsection looks at 2-cocycles in terms of their interpretation in terms of central extensions. It leads to a restatement / rediscovery / different perspective on some of the results of the preceding subsections.

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

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

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 $H^2(G,A)$.

The key insight is as follows:

The 2-cocycle is a normalized 2-cocycle if and only if $s(1)$ (which needs to be an element of $A$ inside $E$) is the identity element of $E$ (and hence also of $A$ as a subgroup of $E$).

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. This is just a restatement of our earlier observation that $Z_n^2(G,A)$ maps surjectively to $H^2(G,A)$.

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.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness	Capture of difference	Intermediate notions
cyclicity-preserving 2-cocycle for trivial group action	A 2-cocycle $f$ such that $f(g,h)=0$ whenever $⟨g,h⟩$ is cyclic				IIP 2-cocycle for trivial group action|FULL LIST, MORE INFO
IIP 2-cocycle for trivial group action	A 2-cocycle $f$ such that $f(g,g^{-1})=0$ for all $g$				|FULL LIST, MORE INFO
normalized 2-coboundary for trivial group action	A 2-coboundary $f$ such that $f(g,1)=f(1,g)=0$ for all $g$			The quotient group is canonically isomorphic to the second cohomology group for trivial group action $H^2(G,A)$	|FULL LIST, MORE INFO
bihomomorphism with both input groups identical (equal to $G$) and the output group abelian (equal to $A$)	A function $f:G\times G\to A$ such that, if either input for $f$ is fixed, $f$ is a homomorphism from the other input to $A$	bihomomorphism to abelian group implies 2-cocycle proves the 2-cocycle part; the normalized part follows directly from the definition of bihomomorphism			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness	Capture of difference	Intermediate notions
2-cocycle for trivial group action				The quotient group (with canonical splitting) is isomorphic to the base group ($A$), with the canonical splitting being constant functions; see #Importance for more	|FULL LIST, MORE INFO