<!
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, 102);"
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: left;">A
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>
