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<td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">CR_ChainMapFrom</span><br
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">
<span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">Cocycle(R,f,p,n)</span></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 R<sub>p</sub>
→ Z and positive integers p, n. It outputs a
function F(w) which gives the image in R<sub>n</sub>, under a chain map
of degree -p induced by f, of a word w in R<sub>n+p</sub>. The
resolution R must have a contracting homotopy.</td>
</tr>
<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255); color: rgb(255, 204, 0);"><span
style="font-family: helvetica,arial,sans-serif; color: rgb(0, 0, 0);">CR_CocyclesAnd<br>
Coboundaries(R,n)<br>
<br>
</span><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">CR_CocyclesAnd<br>
Coboundaries<br>
(R,n,true)<span style="font-family: serif;"><span
style="color: rgb(0, 0, 102);"></span></span></span><br>
</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: <br>
<ul>
<li><span style="font-family: helvetica,arial,sans-serif;">CC.cocyclesBasis</span>
is a basis for
the
abelian group of integral cocycles µ : R<sub>n</sub> → Z. Such a
ZG-homomorphism µ is represented by
the
integer vector v=[µ(e<sub>1</sub>), ..., µ(e<sub>k</sub>)]
where e<sub>i</sub> are the free ZG-generators of R<sub>n</sub>.</li>
<li>Any coboundary ß : R<sub>n</sub> → Z is a linear
combination of basis cocycles and we denote by (ß) the
coefficients in this combination. <span
style="font-family: helvetica,arial,sans-serif;">CC.boundariesCoefficients</span>
is a list [(ß<sub>1</sub>),
..., (ß<sub>m</sub>)] where the ß<sub>i </sub>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 <span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><span
style="font-family: serif;"><span style="color: rgb(0, 0, 102);">
returns a list [C,B,T,P,Q] where<br>
</span></span></span>
<ul>
<li><span style="font-family: helvetica,arial,sans-serif;">CC.torsionCoefficients</span>
is a list of the torsion coefficients of <span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><span
style="font-family: serif;"><span style="color: rgb(0, 0, 102);">H<sup>n</sup>(G,Z).</span></span></span><br>
<span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><span
style="font-family: serif;"><span style="color: rgb(0, 0, 102);"></span></span></span></li>
<li><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><span
style="font-family: serif;"><span style="color: rgb(0, 0, 102);"><span
style="font-family: helvetica,arial,sans-serif;">CC.cocycleToClass(v)</span>
is a function that, given a vector v representing a cocycle,
returns a vector u representing the corresponding element in H<sup>n</sup>(G,Z).
(</span></span></span>
Let a<sub>i</sub> be the i-th canonical generator of the d-generator
abelian group H<sup>n</sup>(G,Z). The cohomology class n<sub>1</sub>a<sub>1</sub>
+ ... +n<sub>d</sub>a<sub>d </sub>is represented by the integer vector
u=(n<sub>1</sub>, ..., n<sub>d</sub>). )<br>
</li>
<li><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><span
style="font-family: serif;"><span style="color: rgb(0, 0, 102);"><span
style="font-family: helvetica,arial,sans-serif;">CC.ClassToCocycle(u)</span>
is function that, given a vector u representing an element
in </span></span></span><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><span
style="font-family: serif;"><span style="color: rgb(0, 0, 102);">H<sup>n</sup>(G,Z),</span></span></span><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><span
style="font-family: serif;"><span style="color: rgb(0, 0, 102);">
returns a vector v representing a corresponding cocycle. <br>
</span></span></span></li>
</ul>
</td>
</tr>
<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"><span
style="font-family: helvetica,arial,sans-serif;">CR_IntegralClassTo<br>
Cocycle(R,u,n)<br>
<br>
</span><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">CR_IntegralClassTo<br>
Cocycle(R,u,n,A)</span><br>
</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 H<sup>n</sup>(R,Z)=H<sup>n</sup>(G,Z).
It returns an integer vector v representing a corresponding cocycle
(i.e. ZG-homomorphism R<sub>n</sub> → Z).<br>
<br>
Let a<sub>i</sub> be the i-th canonical generator of the d-generator
abelian group H<sup>n</sup>(G,Z). The cohomology class n<sub>1</sub>a<sub>1</sub>
+ ... +n<sub>d</sub>a<sub>d </sub>is represented by the integer vector
u=(n<sub>1</sub>, ..., n<sub>d</sub>).<br>
<br>
Let e<sub>i</sub> be the i-th generator of the free ZG-module R<sub>n</sub>.
A ZG-homomorphism µ : R<sub>n</sub> → Z is represented by the
integer vector v=[µ(e<sub>1</sub>), ..., µ(e<sub>k</sub>)]
where k is the ZG-rank of R<sub>n</sub>.<br>
<br>
To save the function from having to calculate the abelian group H<sup>n</sup>(G,Z)
an optional fourth variable can be used, <span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">IntegralClassToCocycle(R,u,n,A)
<span style="color: rgb(0, 0, 102); font-family: serif;">, where
A is the output of the command <span
style="font-family: helvetica,arial,sans-serif;">CocyclesAndCoboundaries(R,n)</span>
.</span></span> </td>
</tr>
<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255); color: rgb(0, 0, 0);"><span
style="font-family: helvetica,arial,sans-serif;">CR_IntegralCocycleTo<br>
Class(R,v,n)<br>
<br>
</span><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">CR_IntegralCocycleTo<br>
Class(R,v,n,A)</span><br>
</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 R<sub>n</sub> → Z). It
returns an integer vector u representing the corresponding cohomology
class in H<sup>n</sup>(R,Z)=H<sup>n</sup>(G,Z). <br>
<br>
To save the function from having to calculate the abelian group H<sup>n</sup>(G,Z)
an optional fourth variable can be used, <span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">IntegralCocycleToClass(R,v,n,A)
<span style="color: rgb(0, 0, 102);"><span
style="font-family: serif;">, where </span></span></span><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;"><span
style="color: rgb(0, 0, 102); font-family: serif;">A is the output of
the command <span style="font-family: helvetica,arial,sans-serif;">CocyclesAndCoboundaries(R,n)</span>
.</span></span> </td>
</tr>
<tr>
<td
style="vertical-align: top; background-color: rgb(255, 255, 255);"><span
style="color: rgb(0, 0, 0); font-family: helvetica,arial,sans-serif;">CR_IntegralCycleTo<br>
Class(R,n)(v)</span><br>
</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.<br>
</td>
</tr>
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distrib
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Mandriva
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2010.0
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i586
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media
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contrib-release
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