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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <meta http-equiv="content-type" content="text/html; charset=ISO-8859-1"> <title>AboutHap</title> </head> <body style="color: rgb(0, 0, 153); background-color: rgb(204, 255, 255);" link="#000066" vlink="#000066" alink="#000066"> <br> <table style="text-align: left; margin-left: auto; margin-right: auto; color: rgb(0, 0, border="0" cellpadding="20" cellspacing="10"> <tbody> <tr align="center"> <th style="vertical-align: top;"> <table style="width: 100%; text-align: left;" cellpadding="2" cellspacing="2"> <tbody> <tr> <td style="vertical-align: top;"><a href="aboutArithmetic.html"><small style="color: rgb(0, 0, 102);">Previous</small></a><br> </td> <td style="text-align: center; vertical-align: top; color: rgb(0, 0, 102);"><big><span style="font-weight: bold;">About HAP: Bredon Homology<br> </span></big></td> <td style="text-align: right; vertical-align: top;"><a href="aboutDavisComplex.html"><small style="color: rgb(0, 0, 102);">next</small></a><br> </td> </tr> </tbody> </table> <big><span style="font-weight: bold;"></span></big><br> </th> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 255); text-align: le G-CW complex X is a CW space with an action of a group G that induces a permutation of cells. The space is said to be <span style="font-style: italic;">rigid</span> if any element of G that stabilizes a cell stabilizes it point-wise. <br> <br> We denote by O<sub>G</sub> the category with one object G/H for each finite subgroup H in G, and with maps G/H --> G/H' the morphisms of G-sets.<br> <br> A Bredon module is a contravariant functor M:O<sub>G</sub> ---> Ab to the category of abelian groups.<br> <br> Standard examples of Bredon modules are:<br> <ul> <li>the contravariant functor M=B that sends an object G/H to the free abelian group BS(H) with isomorphism types of transitive H-sets as basis.</li> <li>the contravariant functor M=R that sends an object G/H to the vector space R<span style="font-weight: bold;"><sub>C</sub></span>(H) of complex representations of the finite group H.</li> </ul> <br> We denote by H<sub>n</sub>(X,M) the Bredon homology of a rigid G-CW space with coefficients in a Bredon module M.<br> <br> The following functions for computing Bredon homology are joint work with <span style="font-weight: bold;">Bui Anh Tuan</span>.<br> </td> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 255);">The following commands compute the Bredon homology H<sub>1</sub>(K,B)=0 of the Quillen complex K(G,p) at the prime p=3 for the symmetric group G=S<sub>9 </sub>with coefficients in the Burnside ring B. The simplicial complex K(G,p) is the order complex of the poset of non-trivial elementary abelian p-subgroups of G. The G-action on K is induced by congugation and is rigid.<br> </td> </tr> <tr> <td style="background-color: rgb(255, 255, 204); vertical-align: top;">gap> G:=SymmetricGroup(9);;<br> gap> K:=QuillenComplex(G,3);<br> Simplicial complex of dimension 2.<br> gap> R:=GChainComplex(K,G);<br> G-chain complex in characteristic 0 for Sym( [ 1 .. 9 ] ) .<br> gap> C:=TensorWithBurnsideRing(R);<br> Chain complex of length 2 in characteristic 0 .<br> gap> Homology(C,1);<br> [ ]<br> </td> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 255);">The following commands compute the the Bredon homology H<sub>0</sub>(<span style="text-decoration: underline;">E</span>SL<sub>3</sub>(Z),R) = Z<sup>8</sup> of a classifying space for proper actions for the special linear group SL<sub>3</sub>(Z); the coefficients are in the complex representation ring R.<br> </td> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 204);">gap> R:=ContractibleGcomplex("SL(3,Z)s");<br> Non-free resolution in characteristic 0 for <matrix group> .<br> <br> gap> D:=TensorWithComplexRepresentationRing(R);<br> Chain complex of length 3 in characteristic 0 .<br> <br> gap> Homology(D,0);<br> [ 0, 0, 0, 0, 0, 0, 0, 0 ]<br> </td> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 255);">The following commands compute the the Bredon homology H<sub>1</sub>(<span style="text-decoration: underline;">E</span>G,R) = Z<sub>2</sub>+Z<sup>3</sup> of a classifying space for proper actions for the crystallographic group G=SpaceGroup(3,32); the coefficients are in the Burnside ring B.</td> </tr> <tr> <td style="vertical-align: top; background-color: rgb(255, 255, 204);">gap> G:=SpaceGroup(3,32);;<br> gap> gens:=GeneratorsOfGroup(G);;<br> gap> bas:=CrystGFullBasis(G);;<br> gap> R:=CrystGcomplex(gens,bas,0);;<br> gap> D:=TensorWithBurnsideRing(R);;<br> gap> Homology(D,1);<br> [ 2, 0, 0, 0 ]</td> </tr> <tr> <td style="vertical-align: top;"> <table style="margin-left: auto; margin-right: auto; width: 100%; text-align: left;" border="0" cellpadding="2" cellspacing="2"> <tbody> <tr> <td style="vertical-align: top;"><a style="color: rgb(0, 0, 102);" href="aboutArithmetic.html">Previous Page</a><br> </td> <td style="text-align: center; vertical-align: top;"><a href="aboutContents.html"><span style="color: rgb(0, 0, 102);">Contents</span></a><br> </td> <td style="text-align: right; vertical-align: top;"><a href="aboutDavisComplex.html"><span style="color: rgb(0, 0, 102);">Next page</span><br> </a> </td> </tr> </tbody> </table> <a href="aboutTopology.html"><br> </a> </td> </tr> </tbody> </table> <br> <br> </body> </html>
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2026-04-02