<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (HAP) - Chapter 13: Cocycles</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> </head> <body><a href="../www/index.html"><small>HAP home</small></a> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap12.html">Previous Chapter</a> <a href="chap14.html">Next Chapter</a> </div> <p><a id="X85A9B66278AF63D9" name="X85A9B66278AF63D9"></a></p> <div class="ChapSects"><a href="chap13.html#X85A9B66278AF63D9">13. <span class="Heading"> Cocycles</span></a> </div> <h3>13. <span class="Heading"> Cocycles</span></h3> <div class="pcenter"><table cellspacing="10" class="GAPDocTable"> <tr> <td class="tdleft"><code class="code"> CcGroup(A,f) </code></p> <p>Inputs a G-module A (i.e. an abelian G-outer group) and a standard 2-cocycle f G x G ---> A. It returns the extension group determined by the cocycle. The group is returned as a CcGroup.</p> <p>This is a HAPcocyclic function and thus only works when HAPcocyclic is loaded.</td> </tr> <tr> <td class="tdleft"><code class="code"> CocycleCondition(R,n) </code></p> <p>Inputs a resolution R and an integer n>0. It returns an integer matrix M with the following property. Suppose d=R.dimension(n). An integer vector f=[f_1, ... , f_d] then represents a ZG-homomorphism R_n --> Z_q which sends the ith generator of R_n to the integer f_i in the trivial ZG-module Z_q (where possibly q=0 ). The homomorphism f is a cocycle if and only if M^tf=0 mod q.</td> </tr> <tr> <td class="tdleft"><code class="code"> StandardCocycle(R,f,n) </code> <br /> <code class="code"> StandardCocycle(R,f,n,q) </code></p> <p>Inputs a ZG-resolution R (with contracting homotopy), a positive integer n and an integer vector f representing an n-cocycle R_n --> Z_q where G acts trivially on Z_q. It is assumed q=0 unless a value for q is entered. The command returns a function F(g_1, ..., g_n) which is the standard cocycle G_n --> Z_q corresponding to f. At present the command is implemented only for n=2 or 3.</td> </tr> <tr> <td class="tdleft"><code class="code"> Syzygy(R,g) </code></p> <p>Inputs a ZG-resolution R (with contracting homotopy) and a list g = [g[1], ..., g[n]] of elements in G. It returns a word w in R_n. The word w is the image of the n-simplex in the standard bar resolution corresponding to the n-tuple g. This function can be used to construct explicit standard n-cocycles. (Currently implemented only for n<4.)</td> </tr> </table><br /><p> </p><br /> </div> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap12.html">Previous Chapter</a> <a href="chap14.html">Next Chapter</a> </div> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>