## The `Crime` Package

Version 1.3
Marcus Bishop marcus.bishop@gmail.com
&copyright; 2006, 2007 Marcus Bishop

Crime is free software which is distributed under the GNU Public Licence, version 2, and may be redistributed under the GNU Public Licence, version 2 or later (at your preference). See the file COPYING for detailed information
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 shared 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-1.3.tar.gz % tar xvzvf crime-1.3.tar.gz In this situation, users would 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, and load the package as follows. $ mkdir /tmp/pkg $ cd /tmp/pkg $ wget http://math.uic.edu/~marcus/Crime/crime-1.3.tar.gz $ tar xvzvf crime-1.3.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 Appendix .

Usage
Unless otherwise specified, all the functions described below taking an argument n do whatever the manual says they do up to homological degree n. 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 CObject.
This function creates a CObject having components the p-group G and the &MeatAxe; module M, which should be a kG-module with G the same group and k a field of characteristic p. Note that &MeatAxe; modules record k but not G, which is why this operation requires the user to specify G but not k.

Fortunately, most users don't need to know anything about &MeatAxe; modules, being interested primarily in the case where k=\mathbb{F}_p, k=GF(p), and M=k, the trivial kG-module. In this situation, the second invocation creates a cohomology object having components the p-group G and the trivial &MeatAxe; kG-module k=\mathbb{F}_p. k=GF(p).

We emphasize that in the first invocation, k can be any field of characteristic p and M can be any &MeatAxe; module over kG, and that ProjectiveResolution works when M is an arbitrary &MeatAxe; module, but that 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 n 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. Because of the minimality of P_\ast,P, the groups \mathrm{Ext}^i_{kG}\left(M,N\right) Ext^i(M,N) are simply \mathrm{Hom}_{kG}\left(P_i,N\right), Hom(P_i,N), and if M and N are both the trivial kG-module k, then H^i\left(G,k\right)=\mathrm{Ext}^i_{kG} \left(k,k\right)=k^{b_i}. H^i(G,k)=Ext^i(k,k)=k^(b_i).

a list containing the Betti numbers b_0, b_1,\ldots, b_n.
Given a cohomology object C having components G and M, this function computes the first n+1 terms of the minimal projective resolution P_\astP 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 b_{n-1}\left|G\right| b_n x b_{n-1}|G| 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
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.

a list containing the degrees of the elements of a set of generators of the cohomology ring.
Given a cohomology object C having group component G and module component the trivial kG-module, this function computes a set of generators of H^\ast\left(G,k\right) H*(G,k) having degree n or less, 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 group component G and module component k, 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 n or less.

More specifically, 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 in section ) and subject to the relators in the second list. See section for more details still.

For example, consider 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 ] ] This tells 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 section 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, you could restart &GAP; and do it again.

Finally, CohomologyRelators is not idempotent for efficiency reasons, so sadly, if you don't uncover all the relators the first time, you will have to start all over from the beginning.
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
Whereas the operations in sections and calculate a presentation for the cohomology ring, the operation below creates the ring in &GAP; as a structure constant algebra.

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 with group component G and module component the trivial kG-module, 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.

Recall that H^\ast\left(G,k\right) H*(G,k) is a graded structure constant algebra, the components being the cohomology groups H^i\left(G,k\right). H^i(G,k). The following functions were intended to be used for cohomology rings, but in principle, they work for any graded structure constant algebra.

true
or false.
Given an element e of a cohomology ring H^\ast\left(G,k\right), H*(G,k), this operation determines whether or not e is homogeneous, that is, whether e is contained in H^i\left(G,k\right) H^i(G,k) for some i.
the degree of e.
This function returns the degree of the possibly non-homogeneous element e of a cohomology ring H^\ast\left(G,k\right). H*(G,k). Specifically, if H^\ast\left(G,k\right)=A_0\oplus A_1\oplus A_2\oplus\cdots H*(G,k) = A_0 + A_1 + A_2 + \ldots where A_i=H^i\left(G,k\right), A_i = H^i(G,k), 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 ). It points out 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_\astP is a minimal projective resolution, we have H^i\left(G,k\right) = \mathrm{Hom}_{kG}\left(P_i,k\right) H^i(G,k) = Hom_{{kG}}(P_i,k) where P_i = \left(kG\right)^{b_i} 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, for 1\le j\le b_i, j=1,2,\ldots,b_i, where b_i is the kG-rank of P_i.

The command CohomologyRing(C,n) forms the vector space whose basis is the concatenation of the natural bases of H^i\left(G,k\right) 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, this means that the function computes the upper left-hand corner of the multiplication table. If \deg x + \deg y > n, the product xy is taken to be zero. Therefore, the ring returned by CohomologyRing is H^\ast\left(G,k\right)\left/J_{>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 natural basis elements of H^1\left(G,k\right) as generators, and for 1<i\le n, i=2\ldots n, taking all natural basis elements of H^i\left(G,k\right) which are not products of lower-degree elements as generators. Therefore, unless you know that there is an n for which there exists a generating set of H^\ast\left(G,k\right) H*(G,k) consisting of elements of degree n or less, 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. The knowledge of such an n is the subject of section .

Similarly, CohomologyRelators proceeds inductively until degree n, returning a list of polynomials which generate the ideal of relators of multidegree n or less. Again, you have to already know how big n should be.

The result of the preceding 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 k\left\langle x_1,x_2,\ldots x_m \right\rangle is the graded polynomial ring over k in the non-commuting variables x_1,x_2,\ldots,x_m, having degrees the numbers in the list returned by CohomologyGenerators, 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 greater than n, then f won't be surjective. Similarly, if there is a relator of multidegree greater than n which is not a consequence of lower degree relators, then f won't be injective.

See section for a discussion on 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.
This function returns the induced homomorphism 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 the 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_n\right\rangle \left\langle xi, x{{i+1}}, \ldots , xj \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 sizes 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), so they are isomorphic as rings. However, the following example shows that \left\langle z,z,z\right\rangle \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
Some Benchmarks
Some of the key subroutines have been improved in version 1.3 of `Crime`, resulting in a significant speedup of many basic calculations.

The numbers in the following table compare the runtimes of the 1.1 and 1.3 versions of the functions listed in the column headers on all the groups of size 16. All functions were executed on a 2.4 GHz AMD64 processor with 12 GB of RAM. The number in the first column of every row is the Small Group Library number of the group used in that row, so the group used in row n is SmallGroup(16,n). The upper number in each box is the runtime for the 1.1 version, whereas the lower number is the runtime for the 1.3 version.

Group
ProjectiveResolution
CohomologyGenerators
CohomologyRelators
1
0:00:00.424
0:00:00.040
0:00:00.092
0:00:00.380
0:00:00.016
0:00:00.012
2
0:00:00.436
0:00:01.472
0:00:05.848
0:00:00.072
0:00:00.116
0:00:00.396
3
0:00:02.752
0:00:11.469
0:00:42.887
0:00:00.352
0:00:01.192
0:00:01.944
4
0:00:00.480
0:00:01.456
0:00:06.320
0:00:00.068
0:00:00.116
0:00:00.316
5
0:00:00.480
0:00:01.136
0:00:06.045
0:00:00.080
0:00:00.096
0:00:00.296
6
0:00:00.204
0:00:00.724
0:00:01.892
0:00:00.036
0:00:00.072
0:00:00.116
7
0:00:00.504
0:00:01.156
0:00:06.140
0:00:00.088
0:00:00.100
0:00:00.344
8
0:00:00.192
0:00:00.704
0:00:01.800
0:00:00.036
0:00:00.064
0:00:00.120
9
0:00:00.060
0:00:00.100
0:00:00.284
0:00:00.020
0:00:00.012
0:00:00.028
10
0:00:10.161
0:00:34.326
0:03:07.104
0:00:01.524
0:00:04.252
0:00:08.089
11
0:00:10.397
0:00:32.354
0:03:07.355
0:00:01.716
0:00:04.584
0:00:08.021
12
0:00:01.184
0:00:04.613
0:00:20.789
0:00:00.192
0:00:00.544
0:00:00.984
13
0:00:01.152
0:00:04.496
0:00:20.990
0:00:00.196
0:00:00.472
0:00:01.072
14
0:03:26.817
0:07:37.960
0:56:39.273
0:00:41.919
0:02:07.399
0:01:54.752

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, calls ProjectiveResolution, CohomologyGenerators, and CohomologyRelators for each group, and prints the results, as well as the runtimes for each operation, to a file like the one shown below. The runtimes in this example have been deleted, having been presented in Appendix . The example below was computed on a 2.4 GHz AMD64 processor with 12 GB of RAM. See 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 ] Generators in degrees: [ 1, 2 ] Relators: [ [ z, y ], [ z^2 ] ] SmallGroup(16,2) Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] Generators in degrees: [ 1, 1, 2, 2 ] Relators: [ [ z, y, x, w ], [ z^2, y^2 ] ] SmallGroup(16,3) Betti Numbers: [ 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36 ] Generators in degrees: [ 1, 1, 2, 2, 2 ] Relators: [ [ z, y, x, w, v ], [ z^2, z*y, z*x, y^2*v+x^2 ] ] SmallGroup(16,4) Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] Generators in degrees: [ 1, 1, 2, 2 ] Relators: [ [ z, y, x, w ], [ z^2, z*y+y^2, y^3 ] ] SmallGroup(16,5) Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] Generators in degrees: [ 1, 1, 2 ] Relators: [ [ z, y, x ], [ z^2 ] ] SmallGroup(16,6) Betti Numbers: [ 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6 ] Generators in degrees: [ 1, 1, 3, 4 ] Relators: [ [ z, y, x, w ], [ z^2, z*y^2, z*x, x^2 ] ] SmallGroup(16,7) Betti Numbers: [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ] Generators in degrees: [ 1, 1, 2 ] Relators: [ [ z, y, x ], [ z*y ] ] SmallGroup(16,8) Betti Numbers: [ 1, 2, 2, 2, 3, 4, 4, 4, 5, 6, 6 ] Generators in degrees: [ 1, 1, 3, 4 ] Relators: [ [ z, y, x, w ], [ z*y, z^3, z*x, y^2*w+x^2 ] ] SmallGroup(16,9) Betti Numbers: [ 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2 ] Generators in degrees: [ 1, 1, 4 ] Relators: [ [ z, y, x ], [ z*y, z^3+y^3, y^4 ] ] SmallGroup(16,10) Betti Numbers: [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ] Generators in degrees: [ 1, 1, 1, 2 ] Relators: [ [ z, y, x, w ], [ z^2 ] ] SmallGroup(16,11) Betti Numbers: [ 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66 ] Generators in degrees: [ 1, 1, 1, 2 ] Relators: [ [ z, y, x, w ], [ z*y ] ] SmallGroup(16,12) Betti Numbers: [ 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17 ] Generators in degrees: [ 1, 1, 1, 4 ] Relators: [ [ z, y, x, w ], [ z^2+z*y+y^2, y^3 ] ] SmallGroup(16,13) Betti Numbers: [ 1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17 ] Generators in degrees: [ 1, 1, 1, 4 ] Relators: [ [ z, y, x, w ], [ z*y+x^2, z*x^2+y*x^2, y^2*x^2+x^4 ] ] SmallGroup(16,14) Betti Numbers: [ 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, 286 ] Generators in degrees: [ 1, 1, 1, 1 ] Relators: [ [ z, y, x, w ], [ ] ]