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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">CR_ChainMapFrom 
 Cocycle(R,f,p,n)</td>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Inputs
at least n+p terms
of a ZG-resolution R, a vector f representing an integer cocycle Rp
→ Z and positive integers p, n. It outputs a
function F(w) which gives the image in Rn, under a chain map
of degree -p induced by f, of a word w in Rn+p. The
resolution R must have a contracting homotopy.</td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); color: rgb(255, 204, 0);">CR_CocyclesAnd 
Coboundaries(R,n) 
 
 CR_CocyclesAnd 
Coboundaries 
(R,n,true) 
 </td>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Inputs
an integer n>0 and at least n+1terms of a ZG-resolution R. It
returns a record CC with the following components. list [C,B] where: 
 <ul>
 <li>CC.cocyclesBasis
is a basis for
the
abelian group of integral cocycles µ : Rn → Z. Such a
ZG-homomorphism µ is represented by
the
integer vector v=[µ(e1), ..., µ(ek)]
where ei are the free ZG-generators of Rn.</li>
 <li>Any coboundary ß : Rn → Z is a linear
combination of basis cocycles and we denote by (ß) the
coefficients in this combination. CC.boundariesCoefficients
is a list [(ß1),
..., (ßm)] where the ßi range over a
basis for the abelian group of integral coboundaries.</li>
 </ul>
The remaining components are all "fail" unless an optional third input
variable is set equal to
"true". In that case the remaining components are as follows. The
command 
returns a list [C,B,T,P,Q] where 
 
 <ul>
 <li>CC.torsionCoefficients
is a list of the torsion coefficients of Hn(G,Z). 
 </li>
 <li>CC.cocycleToClass(v)
is a function that, given a vector v representing a cocycle,
returns a vector u representing the corresponding element in Hn(G,Z).
(
Let ai be the i-th canonical generator of the d-generator
abelian group Hn(G,Z). The cohomology class n1a1
+ ... +ndad is represented by the integer vector
u=(n1, ..., nd). ) 
 </li>
 <li>CC.ClassToCocycle(u)
is function that, given a vector u representing an element
in Hn(G,Z),
returns a vector v representing a corresponding cocycle. 
 </li>
 </ul>
 </td>
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 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);">CR_IntegralClassTo 
Cocycle(R,u,n) 
 
 CR_IntegralClassTo 
Cocycle(R,u,n,A) 
 </td>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Inputs
an integer n>0, at least n+1 terms of a ZG-resolution R and an
integer vector u
representing an element in the cohomology group Hn(R,Z)=Hn(G,Z).
It returns an integer vector v representing a corresponding cocycle
(i.e. ZG-homomorphism Rn → Z). 
 
Let ai be the i-th canonical generator of the d-generator
abelian group Hn(G,Z). The cohomology class n1a1
+ ... +ndad is represented by the integer vector
u=(n1, ..., nd). 
 
Let ei be the i-th generator of the free ZG-module Rn.
A ZG-homomorphism µ : Rn → Z is represented by the
integer vector v=[µ(e1), ..., µ(ek)]
where k is the ZG-rank of Rn. 
 
To save the function from having to calculate the abelian group Hn(G,Z)
an optional fourth variable can be used, IntegralClassToCocycle(R,u,n,A)
 , where
A is the output of the command CocyclesAndCoboundaries(R,n)
. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);">CR_IntegralCocycleTo 
Class(R,v,n) 
 
 CR_IntegralCocycleTo 
Class(R,v,n,A) 
 </td>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Inputs
an integer n>0, at least n+1 terms of a ZG-resolution R and an
integer vector v
representing a cocycle (i.e. ZG-homomorphism Rn → Z). It
returns an integer vector u representing the corresponding cohomology
class in Hn(R,Z)=Hn(G,Z). 
 
To save the function from having to calculate the abelian group Hn(G,Z)
an optional fourth variable can be used, IntegralCocycleToClass(R,v,n,A)
 , where A is the output of
the command CocyclesAndCoboundaries(R,n)
. </td>
 </tr>
 <tr>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">CR_IntegralCycleTo 
Class(R,n)(v) 
 </td>
 <td
style="vertical-align: top; background-color: rgb(255, 255, 255);">Inputs
a ZG-resolution R and an integer n. It returns a function f(v) which
gives the homology class of a cycle v. 
 </td>
 </tr>
 </tbody>
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