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<Chapter Label="Cohomology for pcp-groups">
<Heading>Cohomology for pcp-groups</Heading>

The &GAP; 4 package &Polycyclic; provides methods to compute the
first and second cohomology group for a pcp-group <M>U</M> and a finite
dimensional <M>&ZZ; U</M> or <M>FU</M> module <M>A</M> where <M>F</M> is a finite field. The
algorithm for determining the first cohomology group is outlined in
<Cite Key="Eic00"/>.
<P/>

As a preparation for the cohomology computation, we introduce the
cohomology records. These records provide the technical setup for
our cohomology computations.

<Section Label="Cohomology records">
<Heading>Cohomology records</Heading>

Cohomology records provide the necessary technical setup for the
cohomology computations for polycyclic groups.

<ManSection>
<Func Name="CRRecordByMats" Arg="U, mats"/>
<Description>
creates an external module. Let <A>U</A> be a pcp group which acts via the
list of matrices <A>mats</A> on a vector space of the form <M>&ZZ;^n</M> or <M>\mathbb{F}_p^n</M>.
Then this function creates a record which can be used as input for the
cohomology computations.
</Description>
</ManSection>

<ManSection>
<Func Name="CRRecordBySubgroup" Arg="U, A"/>
<Func Name="CRRecordByPcp" Arg="U, pcp"/>
<Description>
creates an internal module. Let <A>U</A> be a pcp group and let <A>A</A> be a
normal elementary or free abelian normal subgroup of <A>U</A> or let <A>pcp</A>
be a pcp of a normal elementary of free abelian subfactor of <A>U</A>. Then
this function creates a record which can be used as input for the
cohomology computations.
<P/>

The returned cohomology record <A>C</A> contains the following entries:
<List>
<Mark><A>factor</A></Mark>
  <Item>
  a pcp of the acting group. If the module is external, then this is
  <A>Pcp(U)</A>. If the module is internal, then this is <A>Pcp(U, A)</A> or
  <A>Pcp(U, GroupOfPcp(pcp))</A>.
  </Item>
<Mark><A>mats</A>, <A>invs</A> and <A>one</A></Mark>
  <Item>
  the matrix action of <A>factor</A> with acting matrices, their inverses
  and the identity matrix.
  </Item>
<Mark><A>dim</A> and <A>char</A></Mark>
  <Item>
  the dimension and characteristic of the matrices.
  </Item>
<Mark><A>relators</A> and <A>enumrels</A></Mark>
  <Item>
  the right hand sides of the polycyclic relators of <A>factor</A> as
  generator exponents lists and a description for the corresponding
  left hand sides.
  </Item>
<Mark><A>central</A></Mark>
  <Item>
  is true, if the matrices <A>mats</A> are all trivial. This is used
  locally for efficiency reasons.
  </Item>
</List>

And additionally, if <M>C</M> defines an internal module, then it contains:

<List>
<Mark><A>group</A></Mark>
  <Item>
  the original group <A>U</A>.
  </Item>
<Mark><A>normal</A></Mark>
  <Item>
  this is either <A>Pcp(A)</A> or the input <A>pcp</A>.
  </Item>
<Mark><A>extension</A></Mark>
  <Item>
  information on the extension of <A>A</A> by <A>U/A</A>.
  </Item>
</List>
</Description>
</ManSection>

</Section>


<Section Label="Cohomology groups">
<Heading>Cohomology groups</Heading>

Let <M>U</M> be a pcp-group and <M>A</M> a free or elementary abelian pcp-group
and a <M>U</M>-module. By <M>Z^i(U, A)</M> be denote the group of <M>i</M>-th cocycles
and by <M>B^i(U, A)</M> the <M>i</M>-th coboundaries. The factor <M>Z^i(U,A) / B^i(U,A)</M>
is the <M>i</M>-th cohomology group. Since <M>A</M> is elementary or free abelian,
the groups <M>Z^i(U, A)</M> and <M>B^i(U, A)</M> are elementary or free abelian
groups as well.
<P/>

The &Polycyclic; package provides methods to compute first and
second cohomology group for a polycyclic group <A>U</A>.  We write all
involved groups additively and we use an explicit description by bases
for them.  Let <M>C</M> be the cohomology record corresponding to <M>U</M> and
<M>A</M>.
<P/>

Let <M>f_1, \ldots, f_n</M> be the elements in the entry <M>factor</M> of the
cohomology record <M>C</M>. Then we use the following embedding of the
first cocycle group to describe 1-cocycles and 1-coboundaries:
<M>Z^1(U, A) \to A^n : \delta \mapsto (\delta(f_1), \ldots, \delta(f_n))</M>
<P/>

For the second cohomology group we recall that each element of <M>Z^2(U, A)</M>
defines an extension <M>H</M> of <M>A</M> by <M>U</M>. Thus there is a pc-presentation
of <M>H</M> extending the pc-presentation of <M>U</M> given by the record <M>C</M>.
The extended presentation is defined by tails in <M>A</M>; that is, each
relator in the record entry <M>relators</M> is extended by an element of <M>A</M>.
The concatenation of these tails yields a vector in <M>A^l</M> where <M>l</M> is
the length of the record entry <M>relators</M> of <M>C</M>. We use these tail
vectors to describe <M>Z^2(U, A)</M> and <M>B^2(U, A)</M>. Note that this
description is dependent on the chosen presentation in <M>C</M>. However,
the factor <M>Z^2(U, A)/ B^2(U, A)</M> is independent of the chosen presentation.
<P/>

The following functions are available to compute explicitly the first
and second cohomology group as described above.

<ManSection>
<Func Name="OneCoboundariesCR" Arg="C"/>
<Func Name="OneCocyclesCR" Arg="C"/>
<Func Name="TwoCoboundariesCR" Arg="C"/>
<Func Name="TwoCocyclesCR" Arg="C"/>
<Func Name="OneCohomologyCR" Arg="C"/>
<Func Name="TwoCohomologyCR" Arg="C"/>
<Description>
The first four functions return bases of the corresponding group. The
last two functions need to describe a factor of additive abelian groups.
They return the following descriptions for these factors.
<List>
  <Mark><A>gcc</A></Mark>
   <Item>
   the basis of the cocycles of <A>C</A>.
   </Item>
  <Mark><A>gcb</A></Mark>
   <Item>
   the basis of the coboundaries of <A>C</A>.
   </Item>
  <Mark><A>factor</A></Mark>
   <Item>
   a description of the factor of cocycles by coboundaries.
   Usually, it would be most convenient to use additive mappings here.
   However, these are not available in case that <A>A</A> is free abelian
   and thus we use a description of this additive map as record. This
   record contains
   <List>
   <Mark><A>gens</A></Mark>
    <Item>
    a base for the image.
    </Item>
   <Mark><A>rels</A></Mark>
    <Item>
    relative orders for the image.
    </Item>
   <Mark><A>imgs</A></Mark>
    <Item>
    the images for the elements in <A>gcc</A>.
    </Item>
   <Mark><A>prei</A></Mark>
    <Item>
    preimages for the elements in <A>gens</A>.
    </Item>
   <Mark><A>denom</A></Mark>
    <Item>
    the kernel of the map; that is, another basis for <A>gcb</A>.
    </Item>
   </List>
  </Item>
</List>

There is an additional function which can be used to compute the
second cohomology group over an arbitrary finitely generated abelian
group. The finitely generated abelian group should be realized as a
factor of a free abelian group modulo a lattice. The function is
called as
</Description>
</ManSection>

<ManSection>
<Func Name="TwoCohomologyModCR" Arg="C, lat"/>
<Description>
where <A>C</A> is a cohomology record and <A>lat</A> is a basis for a sublattice
of a free abelian module. The output format is the same as for
<C>TwoCohomologyCR</C>.
</Description>
</ManSection>

</Section>


<Section Label="Extended 1-cohomology">
<Heading>Extended 1-cohomology</Heading>

In some cases more information on the first cohomology group is
of interest. In particular, if we have an internal module given and
we want to compute the complements using the first cohomology group,
then we need additional information. This extended version of first
cohomology is obtained by the following functions.

<ManSection>
<Func Name="OneCoboundariesEX" Arg="C"/>
<Description>
returns a record consisting of the entries
<List>
<Mark><A>basis</A></Mark>
  <Item>
  a basis for <M>B^1(U, A) \leq A^n</M>.
  </Item>
<Mark><A>transf</A></Mark>
  <Item>
  There is a derivation mapping from <M>A</M> to <M>B^1(U,A)</M>. This mapping
  is described here as transformation from <M>A</M> to <A>basis</A>.
  </Item>
<Mark><A>fixpts</A></Mark>
  <Item>
  the fixpoints of <M>A</M>. This is also the kernel of the derivation mapping.
  </Item>
</List>
</Description>
</ManSection>

<ManSection>
<Func Name="OneCocyclesEX" Arg="C"/>
<Description>
returns a record consisting of the entries
<List>
<Mark><A>basis</A></Mark>
  <Item>
  a basis for <M>Z^1(U, A) \leq A^n</M>.
  </Item>
<Mark><A>transl</A></Mark>
  <Item>
  a special solution. This is only of interest in case that <M>C</M> is
  an internal module and in this case it gives the translation vector
  in <M>A^n</M> used to obtain complements corresponding to the elements in
  <M>basis</M>. If <M>C</M> is not an internal module, then this vector is
  always the zero vector.
  </Item>
</List>
</Description>
</ManSection>

<ManSection>
<Func Name="OneCohomologyEX" Arg="C"/>
<Description>
returns the combined information on the first cohomology group.
</Description>
</ManSection>

</Section>


<Section Label="Extensions and Complements">
<Heading>Extensions and Complements</Heading>

The natural applications of first and second cohomology group is
the determination of extensions and complements. Let <M>C</M> be a
cohomology record.
<ManSection>
<Func Name=" ComplementCR" Arg="C, c"/>
<Description>
returns the complement corresponding to the 1-cocycle <A>c</A>. In the case
that <A>C</A> is an external module, we construct the split extension of <M>U</M>
with <M>A</M> first and then determine the complement. In the case that <A>C</A>
is an internal module, the vector <A>c</A> must be an element of the affine
space corresponding to the complements as described by <C>OneCocyclesEX</C>.
</Description>
</ManSection>

<ManSection>
<Func Name=" ComplementsCR" Arg="C"/>
<Description>
returns all complements using the correspondence to <M>Z^1(U,A)</M>. Further,
this function returns fail, if <M>Z^1(U,A)</M> is infinite.
</Description>
</ManSection>

<ManSection>
<Func Name=" ComplementClassesCR" Arg="C"/>
<Description>
returns complement classes using the correspondence to <M>H^1(U,A)</M>. Further,
this function returns fail, if <M>H^1(U,A)</M> is infinite.
</Description>
</ManSection>

<ManSection>
<Func Name=" ComplementClassesEfaPcps" Arg="U, N, pcps"/>
<Description>
Let <M>N</M> be a normal subgroup of <M>U</M>. This function returns the complement
classes to <M>N</M> in <M>U</M>. The classes are computed by iteration over the
<M>U</M>-invariant efa series of <M>N</M> described by <A>pcps</A>. If at some stage in
this iteration infinitely many complements are discovered, then the function
returns fail. (Even though there might be only finitely many conjugacy
classes of complements to <M>N</M> in <M>U</M>.)
</Description>
</ManSection>

<ManSection>
<Func Name=" ComplementClasses" Arg="[V,] U, N"/>
<Description>
Let <M>N</M> and <M>U</M> be normal subgroups of <M>V</M> with <M>N \leq U \leq V</M>. This
function attempts to compute the <M>V</M>-conjugacy classes of complements to
<M>N</M> in <M>U</M>. The algorithm proceeds by iteration over a <M>V</M>-invariant
efa series of <M>N</M>. If at some stage in this iteration infinitely many
complements are discovered, then the algorithm returns fail.
</Description>
</ManSection>

<ManSection>
<Func Name="ExtensionCR" Arg="C, c"/>
<Description>
returns the extension corresponding to the 2-cocycle <M>c</M>.
</Description>
</ManSection>

<ManSection>
<Func Name="ExtensionsCR" Arg="C"/>
<Description>
returns all extensions using the correspondence to <M>Z^2(U,A)</M>. Further,
this function returns fail, if <M>Z^2(U,A)</M> is infinite.
</Description>
</ManSection>

<ManSection>
<Func Name="ExtensionClassesCR" Arg="C"/>
<Description>
returns extension classes using the correspondence to <M>H^2(U,A)</M>. Further,
this function returns fail, if <M>H^2(U,A)</M> is infinite.
</Description>
</ManSection>

<ManSection>
<Func Name="SplitExtensionPcpGroup" Arg="U, mats"/>
<Description>
returns the split extension of <A>U</A> by the <M>U</M>-module described by <A>mats</A>.
</Description>
</ManSection>

</Section>


<Section Label="Constructing pcp groups as extensions">
<Heading>Constructing pcp groups as extensions</Heading>

This section contains an example application of the second cohomology
group to the construction of pcp groups as extensions. The following
constructs extensions of the group of upper unitriangular matrices with
its natural lattice.

<Example><![CDATA[
# get the group and its matrix action
gap> G := UnitriangularPcpGroup(3,0);
Pcp-group with orders [ 0, 0, 0 ]
gap> mats := G!.mats;
[ [ [ 1, 1, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ],
  [ [ 1, 0, 0 ], [ 0, 1, 1 ], [ 0, 0, 1 ] ],
  [ [ 1, 0, 1 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ]

# set up the cohomology record
gap> C := CRRecordByMats(G,mats);;

# compute the second cohomology group
gap> cc := TwoCohomologyCR(C);;

# the abelian invariants of H^2(G,M)
gap> cc.factor.rels;
[ 2, 0, 0 ]

# construct an extension which corresponds to a cocycle that has
# infinite image in H^2(G,M)
gap> c := cc.factor.prei[2];
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -1, 1 ]

gap> H := ExtensionCR( C, c);
Pcp-group with orders [ 0, 0, 0, 0, 0, 0 ]

# check that the extension does not split - get the normal subgroup
gap> N := H!.module;
Pcp-group with orders [ 0, 0, 0 ]

# create the interal module
gap> C := CRRecordBySubgroup(H,N);;

# use the complements routine
gap> ComplementClassesCR(C);
[  ]
]]></Example>

</Section>

</Chapter>

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