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### 2. Usage

All the functions described below taking an argument `n` except `CohomologyRing`, `CohomologyRelators` and `InducedHomomorphismOnCohomology` do whatever the manual says they do until some stage `n`, where `n` is normally the homological degree. These functions are idempotent in the sense that called a second time with the same argument `n`, they do nothing, but called with a bigger `n`, they continue computing from where the previous calculations left off.

#### 2.1 Cohomology Objects

The computation of group cohomology involves several calculations, the results of which are reused in later calculations, and are thus collected in an object of type `CObject`, which is created with the following command.

##### 2.1-1 CohomologyObject

`> CohomologyObject`( G, k, M ) | ( operation )

`> CohomologyObject`( G ) | ( operation )

Returns: a cohomology object.

This function creates a cohomology object, initially having components the p-group G, the field k of characteristic p, and the MeatAxe kG-module M. The second invocation creates a cohomology object, initially having components the p-group G, the field GF(p), and the trivial MeatAxe kG-module.

We emphasize that in the first invocation, `k` can be any field of characteristic p and that `M` can be any MeatAxe module over kG. However, since the case k=GF(p), and M=k is probabaly the most common, the second invocation is provided for convenience.

The cohomology object is used to store, in addition to the group, field, and module, the boundary maps, the Betti numbers, the multiplication table, etc.

#### 2.2 Minimal Projective Resolutions

Given a p-group G, the field k of characteristic p and a kG-module M, the function below computes a minimal projective resolution

\[ P_n -> ... P_2 -> P_1 -> P_0 -> M -> 0 \]

where P_i=(kG)^(b_i) for certain numbers b_i, the Betti numbers of the resolution. Then the groups Ext^n(M,N) are simply Hom(P_n,N), and if N=k is the trivial kG-module, then H^n(G,k)=Ext^n(k,k)=k^(b_n).

##### 2.2-1 ProjectiveResolution

`> ProjectiveResolution`( C, n ) | ( operation )

Returns: a list containing the Betti numbers b_0, b_1,..., b_n.

Given a cohomology object `C` having components G, k, and M, this function computes the first `n`+1 terms of the minimal projective resolution P* of M of the form P_i=(kG)^(b_i) for i=0,1,...,n, and returns the numbers b_i as a list.

##### 2.2-2 BoundaryMap

`> BoundaryMap`( C, n ) | ( operation )

Returns: the nth boundary map.

Given the cohomology object `C`, this function computes a projective resolution to degree `n` if it hasn't been computed already, and returns the `n`th boundary map.

The map is returned is a b_n -by- |G|b_n-1 matrix, having in the ith row the image of the element 1_G from the ith direct summand of P_n.

See the file `doc/example.*` for an example of the usage and interpretation of the result of this function.

#### 2.3 Cohomology Generators and Relators

##### 2.3-1 CohomologyGenerators

`> CohomologyGenerators`( C, n ) | ( operation )

Returns: a list containing the degrees of the generators of the cohomology ring.

Given a cohomology object `C` having components G, k, and M, this function computes the generators of H*(G,k) of degree less than or equal to `n`, and stores them in `C`. The function returns a list of the degrees of the generators.

The actual cohomology generators are represented by maps P_n -> k and are stored in `C` as column vectors. Only their degrees are returned.

##### 2.3-2 CohomologyRelators

`> CohomologyRelators`( C, n ) | ( operation )

Returns: a list of generators and a list of relators.

Given a cohomology object `C` having components G, k, and M, this function computes a set of generators of the ideal of relators of H*(G,k) having multidegree less than or equal to n.

The function returns two lists, the first containing the variables `z`, `y`, `x`, ... corresponding to the generators of H*(G,k) if there are fewer than 12 generators, and containing the variables `x_1`, `x_2`, `x_3`, ... otherwise. The second is a list of polynomials in the variables from the first list.

While this isn't likely to occur, we point out that if there are 12 or more generators and some of the indeterminates `x_1`, `x_2`, `x_3`, ... have already been named, say by a previous call to `CohomologyRelators`, then these variables will retain their old names. If this is confusing, restart GAP and do it again.

These two lists should be interpreted as follows. The degree `n` truncation of the cohomology ring H*(G,k) is the polynomial ring in the non-commuting variables from the first list, having the degrees returned by `CohomologyGenerators` above, and subject to the relators in the second list.

For example, the following commands

    gap> C:=CohomologyObject(DihedralGroup(8));
    <object>
    gap> CohomologyGenerators(C,10);
    [ 1, 1, 2 ]
    gap> CohomologyRelators(C,10);
    [ [ z, y, x ], [ z*y+y^2 ] ]

tell us that for G=D_8, the cohomology ring H*(G,k) is the graded-commutative polynomial ring in the variables z, y, and x of degrees 1, 1, and 2, subject to the relation zy+y^2. But since H*(G,k) is commutative, k being of characteristic 2, we have H*(G,k)=k[z,y,x]/(zy+y^2). This result can be further improved by taking z=z+y, giving H*(G,k)=k[z,y,x]/(zy).

#### 2.4 Cohomology Rings

See [CT+03] for the details of the calculation of cohomology products using composition of chain maps. See also the file `doc/explanation.*` for an explanation of the implementation.

##### 2.4-1 CohomologyRing

`> CohomologyRing`( C, n ) | ( operation )

`> CohomologyRing`( G, n ) | ( operation )

Returns: the cohomology ring of G.

Given a cohomology object `C` having module component the trivial kG-module and possibly having a projective resolution already computed, this function returns the degree `n` truncation of the cohomology ring H*(G,k). The object returned is an structure constant algebra.

Users interested only in working with the cohomology ring of a group as a GAP object, and not in calculating generators, relators, induced maps, etc, can use the second invocation of this function, which returns the cohomology ring of the group `G` immediately, throwing away all intermediate calculations.

Observe that the object returned is a degree `n` truncation of the infinite-dimensional cohomology ring. A consequence of this is that multiplying two elements whose product has degree greater than `n` results in zero, whether or not the product is really zero.

Observe also that calling `CohomologyRing` a second time with a bigger `n` does not extend the previous ring, but rather, recalculates the entire ring from the beginning. Extending the previous ring appears not to be worth the effort for technical reasons, since almost everything would need to be recalculated again anyway.

##### 2.4-2 IsHomogeneous

`> IsHomogeneous`( e ) | ( operation )

Returns: `true` or `false`.

Given an element `e` of some cohomology ring A, this operation determines whether or not `e` is homogeneous, that is, whether or not `e` is contained in some `hom_component` of A.

##### 2.4-3 Degree

`> Degree`( e ) | ( method )

Returns: the degree of `e`.

This function is intended to return the degree of the possibly non-homogeneous element `e` of some cohomology ring A, but in principle, works for any element of any graded `SCAlgebra`. Specifically, if A = A_0 + A_1 + A_2 + ... with A_i the `hom_components` of A, then this function returns the minimum n such that `e` is in A_0 + A_1 + ... + A_n.

    gap> A:=CohomologyRing(DihedralGroup(8),10);
    <algebra of dimension 66 over GF(2)>
    gap> b:=Basis(A);
    CanonicalBasis( <algebra of dimension 66 over GF(2)> )
    gap> x:=b[2]+b[4];
    v.2+v.4
    gap> IsHomogeneous(x);
    false
    gap> Degree(x);
    2

##### 2.4-4 LocateGeneratorsInCohomologyRing

`> LocateGeneratorsInCohomologyRing`( C ) | ( function )

Returns: a list containing the cohomology generators.

Having already called `CohomologyRing` (see 2.4-1), this function returns a list of elements of the cohomology ring which together with the identity element generate the cohomology ring.

This function is a wrapper for `CohomologyGenerators` (see 2.3-1), indicating which elements of the cohomology ring correspond with the generators found by `CohomologyGenerators`.

    gap> C:=CohomologyObject(SmallGroup(8,4));
    <object>
    gap> A:=CohomologyRing(C,10);
    <algebra of dimension 17 over GF(2)>
    gap> L:=LocateGeneratorsInCohomologyRing(C);
    [ v.2, v.3, v.7 ]
    gap> A=Subalgebra(A,Concatenation(L,[One(A)]));
    true

#### 2.5 Induced Maps

Let f: H -> G be a group homomorphism. Then f induces a homomorphism on cohomology H*(G,k) -> H*(H,k) which is returned by the following function.

##### 2.5-1 InducedHomomorphismOnCohomology

`> InducedHomomorphismOnCohomology`( C, D, f, n ) | ( function )

Returns: the induced homomorphism on cohomology rings.

This function returns the induced homomorphism on cohomology H*(G,k) -> H*(H,k) where the groups H and G are the components of the cohomology objects `C` and `D` and f: H -> G is a group homomorphism. If the cohomology rings have not yet been calculated, they will be computed to degree n, and in this case, they can then be accessed by calling `CohomologyRing` (see 2.4-1).

##### 2.5-2 Inclusion

`> Inclusion`( H, G ) | ( function )

Returns: the inclusion H-> G

This function returns the group homomorphism H-> G when H is a subgroup of G. The returned map can be used as the `f` argument of `InducedHomomorphismOnCohomology`, in which case the induced homomorphism is the restriction map Res: H*(G,k) -> H*(H,k).

The following example calculates the homomorphism on cohomology induced by the inclusion of the cyclic group of size 4 into the dihedral group of size 8.

    gap> G:=DihedralGroup(8);H:=Subgroup(G,[G.2]);
    <pc group of size 8 with 3 generators>
    Group([ f2 ])
    gap> C:=CohomologyObject(H);D:=CohomologyObject(G);
    <object>
    <object>
    gap> i:=Inclusion(H,G);
    [ f2 ] -> [ f2 ]
    gap> Res:=InducedHomomorphismOnCohomology(C,D,i,10);;
    gap> A:=CohomologyRing(D,10);
    <algebra of dimension 66 over GF(2)>
    gap> LocateGeneratorsInCohomologyRing(D);
    [ v.2, v.3, v.6 ]
    gap> A.1^Res; A.2^Res; A.3^Res; A.6^Res;
    v.1
    0*v.1
    v.2
    v.3

#### 2.6 Massey Products

See [K66] for the definitions and [B01] for the details of the calculation using the Yoneda cocomplex. See also the file `doc/explanation.*` for an explanation of the implementation.

##### 2.6-1 MasseyProduct

`> MasseyProduct`( x1, x2, ..., xn ) | ( function )

Returns: the Massey product < x1, x2, ... , xn>.

Given elements x1, x2, ... , xn of a cohomology ring returned by `CohomologyRing` (see 2.4), this function computes the n-fold Massey product < x1, x2, ... , xn > provided that the lower-degree Massey products < x_i ,x_{i+1}, ... , x_j > vanish for all 1 <= i < j <= n, and returns `fail` otherwise.

As an example, recall that the cohomology rings of the cyclic groups C_3 and C_9 of size 3 and 9 over k=GF(3) are both given by k< z,y >/(z^2), that is, they are isomorphic as rings. However, the following example shows that < z, z, z > is non-zero in H*(C_3,k) but is zero in H*(C_9,k).

    gap> A:=CohomologyRing(CyclicGroup(3),10);
    <algebra of dimension 11 over GF(3)>
    gap> z:=Basis(A)[2];
    v.2
    gap> MasseyProduct(z,z);
    0*v.1
    gap> MasseyProduct(z,z,z);
    v.3
    gap> A:=CohomologyRing(CyclicGroup(9),10);
    <algebra of dimension 11 over GF(3)>
    gap> z:=Basis(A)[2];
    v.2
    gap> MasseyProduct(z,z);
    0*v.1
    gap> MasseyProduct(z,z,z);
    0*v.1
    gap> MasseyProduct(z,z,z,z,z,z,z,z,z);
    v.3

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