<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <meta content="text/html; charset=ISO-8859-1" http-equiv="content-type"> <title>functions</title> </head> <body style="color: rgb(0, 0, 102); background-color: rgb(204, 255, 255);" alink="#000066" link="#000066" vlink="#000066"> <div style="text-align: center;"><small></small></div> <table style="width: 100%; text-align: left;" border="0" cellpadding="10" cellspacing="10"> <tbody> <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_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> </tbody> </table> <br> <br> </body> </html>