File: chap9.txt

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  9 Bredon homology
  
  
  9.1 Davis complex
  
  The following example computes the Bredon homology
  
  underline H_0(W,cal R) = Z^21
  
  for  the  infinite Coxeter group W associated to the Dynkin diagram shown in
  the computation, with coefficients in the complex representation ring.
  
    Example  
    gap> D:=[[1,[2,3]],[2,[3,3]],[3,[4,3]],[4,[5,6]]];;
    gap> CoxeterDiagramDisplay(D);
    
  
  
    Example  
    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) = Z_2⊕ Z^9
  
  underline H_1(SL_2(cal O_-3),cal R) = Z
  
  for  cal  O_-3 the ring of integers of the number field Q(sqrt-3), and cal R
  the complex reflection ring.
  
    Example  
    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) = 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.
  
    Example  
    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 ]