## The `Crime` Package

Version 1.1
Marcus Bishop marcus.bishop@gmail.com
&copyright; 2006 Marcus Bishop

We adopt the copyright regulations of &GAP; as detailed in the copyright notice in the &GAP; manual.
This project would not have been possible without Jon Carlson. Jon devised the algorithms used by ProjectiveResolution, CohomologyGenerators, and CohomologyRelators, having already implemented them in `Magma` and sharing these programs with me.
Installation and Loading
Like other &GAP; packages, you download and unpack this package into &GAP;'s pkg directory. For example, if you were using some Unix derivative and &GAP; were installed in the directory /usr/local/gap4r4, then you would do the following. $ cd /usr/local/gap4r4/pkg $ su % wget 'http://math.uic.edu/~marcus/Crime/crime-1.1.tar.gz' % tar xvzvf crime-1.1.tar.gz In this situation, users would then load the package with the LoadPackage command. $ gap gap> LoadPackage("crime");

Users not having root access, using someone else's computer, or having bad relationships with their network administrators, could install the package into their home directories or into some other writable directory such as /tmp as follows. $ mkdir /tmp/pkg $ cd /tmp/pkg $ wget 'http://math.uic.edu/~marcus/Crime/crime-1.1.tar.gz' $ tar xvzvf crime-1.1.tar.gz $ gap -l ';/tmp' gap> LoadPackage("crime");

Finally, it would be a good idea to run the test file to confirm that all the functions work. gap> ReadPackage("crime","tst/test.g"); You can count yourself lucky if &GAP; doesn't complain about anything. There is also a longer running test file for those having ample free time described in Chapter .

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.

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.

a cohomology object.
This function creates a cohomology object having components the p-group G and the &MeatAxe; kG-module M.

The second invocation creates a cohomology object having components the p-group G and the trivial &MeatAxe; kG-module where k is the field \mathbb{F}_p. GF(p).

We emphasize that in the first invocation, M can be any &MeatAxe; module over kG where k is any field of characteristic p. But since the case k=\mathbb{F}_p, k=GF(p), and M=k is probably the most common, the second invocation is provided for convenience. At the present, ProjectiveResolution works when M is an arbitrary &MeatAxe; module, but all the functions dealing with the ring-structure of H^\ast\left(G,k\right) H*(G,k) require that M be the trivial module.

The cohomology object is used to store, in addition to the items mentioned above, the boundary maps, the Betti numbers, the multiplication table, etc.
Minimal Projective Resolutions

Given a p-group G, a field k of characteristic p and a kG-module M, the function below computes the first few terms of the minimal projective resolution of M P_n\rightarrow\cdots\rightarrow P_2\rightarrow P_1 \rightarrow P_0\rightarrow M\rightarrow 0 where P_i=\left(kG\right)^{\oplus b_i} P_n -> ... P_2 -> P_1 -> P_0 -> M -> 0 where P_i is the direct sum (kG)^(b_i) for certain numbers b_i, the Betti numbers of the resolution. The minimal kG-projective resolution of M is unique up to chain isomorphism. Then the groups \mathrm{Ext}^n_{kG}\left(M,N\right) Ext^n(M,N) are simply \mathrm{Hom}_{kG}\left(P_n,N\right), Hom(P_n,N), and if M=N=k is the trivial kG-module, then H^n\left(G,k\right)=\mathrm{Ext}^n_{kG} \left(k,k\right)=k^{b_n}. H^n(G,k)=Ext^n(k,k)=k^(b_n).

a list containing the Betti numbers b_0, b_1,\ldots, 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_\ast of M of the form P_i=\left(kG\right)^{\oplus b_i} for 0\leq i\leq n, P_i=(kG)^(b_i) for i=0,1,...,n, and returns the numbers b_i as a list.
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 nth boundary map.

The map returned is a b_n\times\left(b_{n-1}\left|G\right|\right) b_n x \left(b_\left(n-1\right)\left|G\right|\right) 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.
Cohomology Generators and Relators
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^\ast\left(G,k\right) 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 these generators.

The actual cohomology generators are represented by maps P_n \rightarrow k and are stored in C as matrices. Only their degrees are returned.

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 in H^\ast\left(G,k\right), H*(G,k), all having multidegree less than or equal to n. Read on for what this means exactly.

The function returns two lists, the first list containing the variables z, y, x, \ldots corresponding to the generators of H^\ast\left(G,k\right) H*(G,k) if there are fewer than 12 generators and containing the variables x&uscore;1, x&uscore;2, x&uscore;3, \ldots otherwise. The second list is a list of polynomials in the variables from the first list.

These two lists should be interpreted as follows. A degree-n approximation of the cohomology ring H^\ast\left(G,k\right) H*(G,k) is given by the polynomial ring over k in the non-commuting variables from the first list, (having degrees given by the list returned by CohomologyGenerators above) and subject to the relators in the second list. See for more details still.

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^\ast\left(G,k\right) 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^\ast\left(G,k\right) H*(G,k) is commutative, k being of characteristic 2, we have H^\ast\left(G,k\right)=k\left[z,y,x\right] \left/\left(zy+y^2\right)\right.. H*(G,k)=k[z,y,x]/(zy+y^2). This result can be further improved by taking z=z+y, giving H^\ast\left(G,k\right)=k\left[z,y,x\right] \left/\left(zy\right)\right.. H*(G,k)=k[z,y,x]/(zy).

Observe that in this case, we knew in advance that there was a set of generators for H^\ast\left(G,k\right) H*(G,k) all having degree less than 10, and that there was a set of generators of the ideal of relators all having multidegree less than 10. See see for details.

While this isn't likely to occur, we point out that if there are 12 or more generators and some of the indeterminates x&uscore;1, x&uscore;2, x&uscore;3, \ldots 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.
Tests for Completion
A test or series of tests for completion of the calculation will hopefully be implemented soon. See for the details.
Cohomology Rings
See 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.

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^\ast\left(G,k\right). H*(G,k). See for what this means exactly.

The object returned is a 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.
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&uscore;component of A.
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\oplus A_1\oplus A_2\oplus\cdots A = A_0 + A_1 + A_2 + \ldots where A_i are the hom&uscore;components of A, then this function returns the minimum n such that e is in A_0\oplus A_1\oplus\cdots\oplus A_n. A_0 + A_1 + \ldots + 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
a list containing the cohomology generators.
Having already called CohomologyRing (see ), 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 ), 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
What Happens if n Isn't Big Enough?
Since P_\ast is a minimal resolution, the cohomology group H^i\left(G,k\right) is the dual of P_n, so that H^i\left(G,k\right) has a natural basis consisting of the maps sending the element 1_G of the jth direct summand of P_i to 1_k and all other direct summands to 0_k for 1\le j\le b_i. j=1,2,\ldots,b_i.

The command CohomologyRing(C,n) concatenates these bases for 1\le i\le n i=1,2,\ldots,n and computes all products of basis elements x and y for which \deg x+\deg y \leq n. Thinking of H^\ast\left(G,k\right) H*(G,k) in terms of it's multiplication table, then this means that the function computes the upper left-hand corner of the multiplication table. If \deg x + \deg y > n then the product xy is taken to be zero. Therefore, the ring returned by CohomologyGenerators is H^\ast\left(G,k\right)\left/J_{&tgt;n}\right. H*(G,k)/J where J_{>n} J is the ideal of all elements of degree >n.

The ring determined by CohomologyGenerators and CohomologyRelators is somewhat different. CohomologyGenerators proceeds inductively, taking all standard basis elements of H^1\left(G,k\right) as generators, and for 1&tlt;i\le n, i=2\ldots n, taking all standard basis elements of H^i\left(G,k\right) which are not products of lower-degree elements as generators. Therefore, unless you have some reason to believe that there exists a generating set for H^\ast\left(G,k\right) H*(G,k) consisting of elements of degree \leq n, then you are not guaranteed that the elements returned by the CohomologyGenerators generate H^\ast\left(G,k\right) H*(G,k) as a ring.

Similarly, CohomologyRelators proceeds inductively until degree n, returning a list of polynomials of multidegree \leq n.

The impact of the preceeding information is that there is a homomorphism f:k\left.\left\langle x_1,x_2,\dots x_m \right\rangle\right./I \to H^\ast\left(G,k\right) k\left\langle x_1,x_2,\ldots, x_m \right\rangle/ I \to H*(G,k) where x_1,x_2,\ldots,x_m represent the elements returned by CohomologyGenerators(C,n), k\left\langle x_1,x_2,\ldots x_m \right\rangle is the polynomial ring over k in the non-commuting variables x_1,x_2,\ldots,x_m, and I is the ideal in k\left\langle x_1,x_2,\ldots, x_m \right\rangle generated by the elements returned by CohomologyRelators(C,n).

Therefore, if there is a generator of degree >n, then f won't be surjective. If there is a relator of multidegree >n which is not a consequence of lower degree relators, then f won't be injective.

See for how big n needs to be to ensure that f be an isomorphism.

Induced Maps
Let f: H \rightarrow G be a group homomorphism. Then f induces a homomorphism on cohomology H^\ast\left(G,k\right) \to H^\ast\left(H,k\right) H\ast(G,k) \rightarrow H\ast(H,k) which is returned by the following function.

the induced homomorphism on cohomology rings.
This function returns the induced homomorphism on cohomology H^\ast\left(G,k\right) \to H^\ast\left(H,k\right) H\ast(G,k) \rightarrow H\ast(H,k) where the groups H and G are the components of the cohomology objects C and D and f: H \rightarrow 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 ).
the inclusion H\rightarrow G
This function returns the group homomorphism H\rightarrow 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 \mathrm{Res}^G_H:H^\ast\left(G,k\right) \to H^\ast\left(H,k\right). Res: H\ast(G,k) \rightarrow H\ast(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
Massey Products
See for the definitions and for the details of the calculation using the Yoneda cocomplex. See also the file doc/explanation.* for an explanation of the implementation. the Massey product \left\langle x_1,x_2,\dots,x_n\right\rangle. \left\langle x1, x2, ... , xn\right\rangle. Given elements x_1,x_2,\dots,x_n x1, x2, \ldots , xn of a cohomology ring returned by CohomologyRing (see ), this function computes the n-fold Massey product \left\langle x_1,x_2,\ldots,x_n\right\rangle \left\langle x1, x2, \ldots , xn \right\rangle provided that the lower-degree Massey products \left\langle x_i ,x_{{i+1}}, \ldots , x_j \right\rangle vanish for all 1 \leq i < j \leq 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=\mathbb{F}_3 k=GF(3) are both given by k\left\langle z, y\right\rangle\left/\left(z^2\right)\right., k\left\langle z,y \right\rangle/\left(z^2\right), that is, they are isomorphic as rings. However, the following example shows that \left\langle z, z, z \right\rangle is non-zero in H^\ast\left(C_3,k\right) H*(C_3,k) but is zero in H^\ast\left(C_9,k\right). 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
Leisure and Recreation: Cohomology Rings of all Groups of Size 16
Below is the output of the test file tst/batch.g. The file runs through all groups of size n, which is initially set to 16, and runs ProjectiveResolution, CohomologyGenerators and CohomologyRelators for each group, and prints the results as well as the timings for each operation to a file. The output below was computed on a 3.06 GHz Intel processor with 3.71 GB of RAM. The projective resolutions are calculated initially to degree 10 and the generators and relators to degree 6, due to the fact that I already knew all the generators and relators to be of degree less than 6, see http://www.math.uga.edu/~lvalero/cohointro.html. See also the file tst/README for suggestions on dealing with other users when running long-running batch processes. SmallGroup(16,1) Betti Numbers: [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] Time: 0:00:05.864 Generators in degrees: [ 1, 2 ] Time: 0:00:00.086 Relators: [ [ z, y ], [ z^2 ] ] Time: 0:00:00.245 SmallGroup(16,2) Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] Time: 0:00:00.931 Generators in degrees: [ 1, 1, 2, 2 ] Time: 0:00:02.874 Relators: [ [ z, y, x, w ], [ z^2, y^2 ] ] Time: 0:00:12.227 SmallGroup(16,3) Betti Numbers: [ 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36 ] Time: 0:00:05.292 Generators in degrees: [ 1, 1, 2, 2, 2 ] Time: 0:00:21.770 Relators: [ [ z, y, x, w, v ], [ z^2, z*y, z*x, y^2*v+x^2 ] ] Time: 0:01:26.166 SmallGroup(16,4) Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] Time: 0:00:01.047 Generators in degrees: [ 1, 1, 2, 2 ] Time: 0:00:03.253 Relators: [ [ z, y, x, w ], [ z^2, z*y+y^2, y^3 ] ] Time: 0:00:14.294 SmallGroup(16,5) Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] Time: 0:00:01.065 Generators in degrees: [ 1, 1, 2 ] Time: 0:00:02.493 Relators: [ [ z, y, x ], [ z^2 ] ] Time: 0:00:13.573 SmallGroup(16,6) Betti Numbers: [ 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6 ] Time: 0:00:00.446 Generators in degrees: [ 1, 1, 3, 4 ] Time: 0:00:01.566 Relators: [ [ z, y, x, w ], [ z^2, z*y^2, z*x, x^2 ] ] Time: 0:00:04.132 SmallGroup(16,7) Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] Time: 0:00:01.076 Generators in degrees: [ 1, 1, 2 ] Time: 0:00:02.495 Relators: [ [ z, y, x ], [ z*y ] ] Time: 0:00:13.862 SmallGroup(16,8) Betti Numbers: [ 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6 ] Time: 0:00:00.465 Generators in degrees: [ 1, 1, 3, 4 ] Time: 0:00:01.570 Relators: [ [ z, y, x, w ], [ z*y, z^3, z*x, y^2*w+x^2 ] ] Time: 0:00:04.350 SmallGroup(16,9) Betti Numbers: [ 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2 ] Time: 0:00:00.140 Generators in degrees: [ 1, 1, 4 ] Time: 0:00:00.255 Relators: [ [ z, y, x ], [ z*y, z^3+y^3, y^4 ] ] Time: 0:00:00.718 SmallGroup(16,10) Betti Numbers: [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ] Time: 0:00:20.139 Generators in degrees: [ 1, 1, 1, 2 ] Time: 0:01:04.158 Relators: [ [ z, y, x, w ], [ z^2 ] ] Time: 0:06:27.688 SmallGroup(16,11) Betti Numbers: [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ] Time: 0:00:20.428 Generators in degrees: [ 1, 1, 1, 2 ] Time: 0:01:04.678 Relators: [ [ z, y, x, w ], [ z*y ] ] Time: 0:06:33.808 SmallGroup(16,12) Betti Numbers: [ 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17 ] Time: 0:00:02.438 Generators in degrees: [ 1, 1, 1, 4 ] Time: 0:00:08.927 Relators: [ [ z, y, x, w ], [ z^2+z*y+y^2, y^3 ] ] Time: 0:00:44.464 SmallGroup(16,13) Betti Numbers: [ 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17 ] Time: 0:00:02.389 Generators in degrees: [ 1, 1, 1, 4 ] Time: 0:00:09.247 Relators: [ [ z, y, x, w ], [ z*y+x^2, z*x^2+y*x^2, y^2*x^2+x^4 ] ] Time: 0:00:44.323 SmallGroup(16,14) Betti Numbers: [ 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286 ] Time: 0:07:00.973 Generators in degrees: [ 1, 1, 1, 1 ] Time: 0:15:40.874 Relators: [ [ z, y, x, w ], [ ] ] Time: 1:54:28.052 Total time: 2:38:14.841