  
  [1X3 [33X[0;0YExamples[133X[101X
  
  [33X[0;0YAlthough  there  are some small examples embedded in chapter [14X4[114X, we will give
  some complete examples in this chapter. Most of these could easily be called
  with  the example script mentioned in chapter [14X2[114X, but we will do them step by
  step for best reproducability.[133X
  
  
  [1X3.1 [33X[0;0YExample 1: Klein Bottle[133X[101X
  
  [33X[0;0YSuppose  we  want to calculate the cohomology of the Klein Bottle. First, we
  need a triangulation. It could look like this:[133X
  
  [4X[32X[104X
    [4X1--2--3--1[104X
    [4X|\ |\ |\ |[104X
    [4X| \| \| \|[104X
    [4X4--5--6--4[104X
    [4X|\ |\ |\ |[104X
    [4X| \| \| \|[104X
    [4X7--8--9--7[104X
    [4X|\ |\ |\ |[104X
    [4X| \| \| \|[104X
    [4X1--3--2--1[104X
  [4X[32X[104X
  
  [33X[0;0YThis results in the following list of maximum simplices:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM := [ [1,2,4], [1,2,7], [1,3,6], [1,3,8], [1,4,6], [1,7,8],[127X[104X
    [4X[25X>[125X [27X[2,3,5], [2,3,9], [2,4,5], [2,7,9], [3,5,6], [3,8,9],[127X[104X
    [4X[25X>[125X [27X[4,5,7], [4,6,9], [4,7,9], [5,6,8], [5,7,8], [6,8,9] ];;[127X[104X
  [4X[32X[104X
  
  [33X[0;0YAs  there  is  no  isotropy and therefore no [22Xμ[122X-map, we can continue with the
  orbifold triangulation and simplicial set:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xot := OrbifoldTriangulation( M, "Klein Bottle" );[127X[104X
    [4X[28X<OrbifoldTriangulation "Klein Bottle" of dimension 2. 18 simplices on 9 vertic\[128X[104X
    [4X[28Xes without Isotropy>[128X[104X
    [4X[25Xgap>[125X [27Xss := SimplicialSet( ot );[127X[104X
    [4X[28X<The simplicial set of the orbifold triangulation "Klein Bottle", computed up \[128X[104X
    [4X[28Xto dimension 0 with Length vector [ 18 ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNow  we  will  need a [5Xhomalg[105X ring. As this is a small example we can compute
  directly  over  ℤ,  so  we  can  use  [5XGAP[105X.  In  case you have [5XRingsForHomalg[105X
  installed you might want to try computing in another computer algebra system
  with  the  command  [10XHomalgRingOfIntegersInCAS()[110X,  replacing  "CAS"  with the
  corresponding system.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XR := HomalgRingOfIntegers();[127X[104X
    [4X[28XZ[128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe  are  almost  there.  Let  us create some coboundary matrices and compute
  their cohomology:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xc := CreateCoboundaryMatrices( ss, 4, R );;[127X[104X
    [4X[25Xgap>[125X [27XC := Cohomology( c, R );[127X[104X
    [4X[28X----------------------------------------------->>>>  Z^(1 x 1)[128X[104X
    [4X[28X----------------------------------------------->>>>  Z^(1 x 1)[128X[104X
    [4X[28X----------------------------------------------->>>>  Z/< 2 >[128X[104X
    [4X[28X----------------------------------------------->>>>  0[128X[104X
    [4X[28X----------------------------------------------->>>>  0[128X[104X
    [4X[28X<A graded cohomology object consisting of 5 left modules at degrees[128X[104X
    [4X[28X[ 0 .. 4 ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis is the cohomology of the Klein Bottle.[133X
  
  
  [1X3.2 [33X[0;0YExample 2: [22XV_4[122X[101X[1X[133X[101X
  
  [33X[0;0Y[5XSCO[105X  can  also  be  used  to compute group cohomology, as every group can be
  represented  as an orbifold with just a single point. For [22XV_4[122X, it would look
  like this:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM := [ [1] ];;[127X[104X
    [4X[25Xgap>[125X [27XV4 := Group( (1,2), (3,4) );;[127X[104X
    [4X[25Xgap>[125X [27Xiso := rec( 1 := V4 );;[127X[104X
    [4X[25Xgap>[125X [27Xot := OrbifoldTriangulation( M, iso, "V4" );[127X[104X
    [4X[28X<OrbifoldTriangulation "V4" of dimension 0. 1 simplex on 1 vertex with Isotrop\[128X[104X
    [4X[28Xy on 1 vertex>[128X[104X
    [4X[25Xgap>[125X [27Xss := SimplicialSet( ot );[127X[104X
    [4X[28X<The simplicial set of the orbifold triangulation "V4", computed up to dimensi\[128X[104X
    [4X[28Xon 0 with Length vector [ 1 ]>[128X[104X
    [4X[25Xgap>[125X [27XR := HomalgRingOfIntegers();[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xc := CreateCoboundaryMatrices( ss, 4, R );;[127X[104X
    [4X[25Xgap>[125X [27XC := Cohomology( c, R );[127X[104X
    [4X[28X----------------------------------------------->>>>  Z^(1 x 1)[128X[104X
    [4X[28X----------------------------------------------->>>>  0[128X[104X
    [4X[28X----------------------------------------------->>>>  Z/< 2 > + Z/< 2 >[128X[104X
    [4X[28X----------------------------------------------->>>>  Z/< 2 >[128X[104X
    [4X[28X----------------------------------------------->>>>  Z/< 2 > + Z/< 2 > + Z/< 2\[128X[104X
    [4X[28X >[128X[104X
    [4X[28X<A graded cohomology object consisting of 5 left modules at degrees[128X[104X
    [4X[28X[ 0 .. 4 ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis is the [22XV_4[122X group cohomology up to degree 4.[133X
  
  
  [1X3.3 [33X[0;0YExample 3: The "Teardrop" orbifold[133X[101X
  
  [33X[0;0YYou  have  seen  a manifold in example 1, and group cohomology in example 2.
  Now  we  will  meet  our  first  proper  orbifold, the Teardrop. This is the
  example  Moerdijk  and  Pronk used in their paper [MP99] on which my work is
  based.  It  is  an  easy  example, but includes both nontrivial isotropy and
  [22Xμ[122X-maps.  We  take the isotropy at the top to be [22XC_2[122X. The triangulation looks
  like this, with the gluing being at [1,3].[133X
  
  [4X[32X[104X
    [4X   3=====1=====3[104X
    [4X  / \   / \   / \[104X
    [4X /   \ /   \ /   \[104X
    [4X5-----2-----4-----5[104X
    [4X       \   /[104X
    [4X        \ /[104X
    [4X         5[104X
  [4X[32X[104X
  
  [33X[0;0YThe source code:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XM := [ [1,2,3], [1,2,4], [1,3,4], [2,3,5], [2,4,5], [3,4,5] ];;[127X[104X
    [4X[25Xgap>[125X [27Xiso := rec( 1 := Group( (1,2) ) );;[127X[104X
    [4X[25Xgap>[125X [27Xmu := [[127X[104X
    [4X[25X>[125X [27X           [ [3], [1,3], [1,2,3], [1,3,4], x -> (1,2) ],[127X[104X
    [4X[25X>[125X [27X           [ [3], [1,3], [1,3,4], [1,2,3], x -> (1,2) ][127X[104X
    [4X[25X>[125X [27X         ];;[127X[104X
    [4X[25Xgap>[125X [27Xot := OrbifoldTriangulation( M, iso, mu, "Teardrop" );[127X[104X
    [4X[28X<OrbifoldTriangulation "Teardrop" of dimension 2. 6 simplices on 5 vertices wi\[128X[104X
    [4X[28Xth Isotropy on 1 vertex and nontrivial mu-maps>[128X[104X
    [4X[25Xgap>[125X [27Xss := SimplicialSet( ot );[127X[104X
    [4X[28X<The simplicial set of the orbifold triangulation "Teardrop", computed up to d\[128X[104X
    [4X[28Ximension 0 with Length vector [ 6 ]>[128X[104X
    [4X[25Xgap>[125X [27XR := HomalgRingOfIntegers();[127X[104X
    [4X[28XZ[128X[104X
    [4X[25Xgap>[125X [27Xc := CreateCoboundaryMatrices( ss, 6, R );;[127X[104X
    [4X[25Xgap>[125X [27XC := Cohomology( c, R );[127X[104X
    [4X[28X----------------------------------------------->>>>  Z^(1 x 1)[128X[104X
    [4X[28X----------------------------------------------->>>>  0[128X[104X
    [4X[28X----------------------------------------------->>>>  Z^(1 x 1)[128X[104X
    [4X[28X----------------------------------------------->>>>  0[128X[104X
    [4X[28X----------------------------------------------->>>>  Z/< 2 >[128X[104X
    [4X[28X----------------------------------------------->>>>  0[128X[104X
    [4X[28X----------------------------------------------->>>>  Z/< 2 >[128X[104X
    [4X[28X<A graded cohomology object consisting of 7 left modules at degrees[128X[104X
    [4X[28X[ 0 .. 6 ]>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis is the Teardrop cohomology.[133X
  
