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9 Bredon homology

9.1 Davis complex

The following example computes the Bredon homology

\(\underline H_0(W,{\cal R}) = \mathbb Z^{21}\)

for the infinite Coxeter group \(W\) associated to the Dynkin diagram shown in the computation, with coefficients in the complex representation ring.

gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,6]]];;
gap> CoxeterDiagramDisplay(D);

gap> C:=DavisComplex(D);;
gap> D:=TensorWithComplexRepresentationRing(C);;
gap> Homology(D,0);
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

9.2 Arithmetic groups

The following example computes the Bredon homology

\(\underline H_0(SL_2({\cal O}_{-3}),{\cal R}) = \mathbb Z_2\oplus \mathbb Z^{9}\)

\(\underline H_1(SL_2({\cal O}_{-3}),{\cal R}) = \mathbb Z\)

for \({\cal O}_{-3}\) the ring of integers of the number field \(\mathbb Q(\sqrt{-3})\), and \(\cal R\) the complex reflection ring.

gap> R:=ContractibleGcomplex("SL(2,O-3)");;
gap> IsRigid(R);
false
gap> S:=BaryCentricSubdivision(R);;
gap> IsRigid(S);
true
gap> C:=TensorWithComplexRepresentationRing(S);;
gap> Homology(C,0);
[ 2, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]
gap> Homology(C,1);
[ 0 ]

9.3 Crystallographic groups

The following example computes the Bredon homology

\(\underline H_0(G,{\cal R}) = \mathbb Z^{17}\)

for \(G\) the second crystallographic group of dimension \(4\) in GAP's library of crystallographic groups, and for \(\cal R\) the Burnside ring.

gap> G:=SpaceGroup(4,2);;
gap> gens:=GeneratorsOfGroup(G);;
gap> B:=CrystGFullBasis(G);;
gap> R:=CrystGcomplex(gens,B,1);;
gap> IsRigid(R);
false
gap> S:=CrystGcomplex(gens,B,0);;
gap> IsRigid(S);
true
gap> D:=TensorWithBurnsideRing(S);;
gap> Homology(D,0);
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ]

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