Normalized 2-cocycle for trivial group action

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Definition

Equivalent definitions in tabular format

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


No. Shorthand Full statement
1 2-cocycle for trivial group action that is zero along the axes is a 2-cocycle for trivial group action: and .
2 2-cocycle for trivial group action that is zero at origin is a 2-cocycle for trivial group action: and .
3 System of coset representatives where identity is represented by identity corresponds to a system of coset representatives for a central extension (corresponding to an element of , 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 , 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 .

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 , we can obtain a normalized 2-cocycle defined as follows:

satisfies the identity for a 2-cocycle because it's just taking the original identity and subtracting twice from both sides.

is also normalized, because and , which are both zero based on 2-cocycle for trivial group action is constant on axes.

Moreover, the difference of and is a constant function with constant value , which is therefore also a 2-coboundary (of the constant function taking value . Thus, belongs to the same cohomology class as , i.e., they project to the same element of the second cohomology group for trivial group action .

Alternative normalization

Given a 2-cocycle for trivial group action , we can obtain a normalized 2-cocycle defined as follows:

  • if are all non-identity elements
  • for all
  • for all (note that it follows from the definition of 2-cocycle that for all , so this can also be written as )

is different from . As the later discussion will clarify, they differ by a 2-coboundary for trivial group action.

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

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

  • is the group of all 2-cocycles for trivial group action.
  • is the subgroup comprising normalized 2-cocycles for trivial group action.
  • is identified with the subgroup comprising constant functions . Specifically, an element corresponds to the constant function sending everything in to .

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

  • is a subgroup of : This is obvious from the definition.
  • (identified as constant functions) is a subgroup of : This is also obvious from the definition of 2-cocycle.
  • : For a normalized 2-cocycle, . If is also constant, this forces it to be zero everywhere.
  • , 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:

  • (where the group in the denominator is realized as constant functions)
  • is a canonically split short exact sequence.
  • is a canonically split short exact sequence.

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

Let's articulate the notation:

  • is a group of 2-cocycles for trivial group action.
  • is the subgroup comprising normalized 2-cocycles.
  • is the subgroup comprising 2-coboundaries.
  • is the subgroup comprising normalized 2-coboundaries; it is the intersection .
  • is identified with constant functions .
  • is the quotient 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:

The new fact of interest is that all constant functions are also 2-coboundaries! That's because the constant function taking value can be realized as the coboundary of the constant function taking value . This means that maps injectively into . This allows us to make a downward arrow from to that is injective. We can also make a downward arrow between the middle groups that is the identity map. This in turn allows us to make a downward map from to , 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 , so we obtain this short exact sequence:

We also have a canonical internal direct sum splitting:

This uses the same logic as the splitting for 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:

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 be a group with a central subgroup isomorphic to (and explicitly identified with) , and a quotient isomorphic to (and explicitly identified with) , 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 be a system of coset representatives for in with being the representation map. Then, define such that

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

Such an is a 2-cocycle for trivial group action of on . 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 .

The key insight is as follows:

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

Since it's always possible to choose so that is the identity element of , 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 can be represented by a normalized 2-cocycle. This is just a restatement of our earlier observation that maps surjectively to .

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 (left and right doesn't matter because is central).
  • The alternative normalization corresponds to keeping all coset representatives intact except , 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 such that whenever is cyclic IIP 2-cocycle for trivial group action|FULL LIST, MORE INFO
IIP 2-cocycle for trivial group action A 2-cocycle such that for all |FULL LIST, MORE INFO
normalized 2-coboundary for trivial group action A 2-coboundary such that for all The quotient group is canonically isomorphic to the second cohomology group for trivial group action |FULL LIST, MORE INFO
bihomomorphism with both input groups identical (equal to ) and the output group abelian (equal to ) A function such that, if either input for is fixed, is a homomorphism from the other input to 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 (), with the canonical splitting being constant functions; see #Importance for more |FULL LIST, MORE INFO