chap3.txt

  
  3 Examples
  
  Although  there  are some small examples embedded in chapter 4, we will give
  some complete examples in this chapter. Most of these could easily be called
  with  the example script mentioned in chapter 2, but we will do them step by
  step for best reproducability.
  
  
  3.1 Example 1: Klein Bottle
  
  Suppose  we  want to calculate the cohomology of the Klein Bottle. First, we
  need a triangulation. It could look like this:
  
  
    1--2--3--1
    |\ |\ |\ |
    | \| \| \|
    4--5--6--4
    |\ |\ |\ |
    | \| \| \|
    7--8--9--7
    |\ |\ |\ |
    | \| \| \|
    1--3--2--1
  
  
  This results in the following list of maximum simplices:
  
    Example  
    gap> M := [ [1,2,4], [1,2,7], [1,3,6], [1,3,8], [1,4,6], [1,7,8],
    > [2,3,5], [2,3,9], [2,4,5], [2,7,9], [3,5,6], [3,8,9],
    > [4,5,7], [4,6,9], [4,7,9], [5,6,8], [5,7,8], [6,8,9] ];;
  
  
  As  there  is  no  isotropy and therefore no μ-map, we can continue with the
  orbifold triangulation and simplicial set:
  
    Example  
    gap> ot := OrbifoldTriangulation( M, "Klein Bottle" );
    <OrbifoldTriangulation "Klein Bottle" of dimension 2. 18 simplices on 9 vertic\
    es without Isotropy>
    gap> ss := SimplicialSet( ot );
    <The simplicial set of the orbifold triangulation "Klein Bottle", computed up \
    to dimension 0 with Length vector [ 18 ]>
  
  
  Now  we  will  need a homalg ring. As this is a small example we can compute
  directly  over  ℤ,  so  we  can  use  GAP.  In  case you have RingsForHomalg
  installed you might want to try computing in another computer algebra system
  with  the  command  HomalgRingOfIntegersInCAS(),  replacing  "CAS"  with the
  corresponding system.
  
    Example  
    gap> R := HomalgRingOfIntegers();
    Z
  
  
  We  are  almost  there.  Let  us create some coboundary matrices and compute
  their cohomology:
  
    Example  
    gap> c := CreateCoboundaryMatrices( ss, 4, R );;
    gap> C := Cohomology( c, R );
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  Z/< 2 >
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  0
    <A graded cohomology object consisting of 5 left modules at degrees
    [ 0 .. 4 ]>
  
  
  This is the cohomology of the Klein Bottle.
  
  
  3.2 Example 2: V_4
  
  SCO  can  also  be  used  to compute group cohomology, as every group can be
  represented  as an orbifold with just a single point. For V_4, it would look
  like this:
  
    Example  
    gap> M := [ [1] ];;
    gap> V4 := Group( (1,2), (3,4) );;
    gap> iso := rec( 1 := V4 );;
    gap> ot := OrbifoldTriangulation( M, iso, "V4" );
    <OrbifoldTriangulation "V4" of dimension 0. 1 simplex on 1 vertex with Isotrop\
    y on 1 vertex>
    gap> ss := SimplicialSet( ot );
    <The simplicial set of the orbifold triangulation "V4", computed up to dimensi\
    on 0 with Length vector [ 1 ]>
    gap> R := HomalgRingOfIntegers();
    Z
    gap> c := CreateCoboundaryMatrices( ss, 4, R );;
    gap> C := Cohomology( c, R );
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  Z/< 2 > + Z/< 2 >
    ----------------------------------------------->>>>  Z/< 2 >
    ----------------------------------------------->>>>  Z/< 2 > + Z/< 2 > + Z/< 2\
     >
    <A graded cohomology object consisting of 5 left modules at degrees
    [ 0 .. 4 ]>
  
  
  This is the V_4 group cohomology up to degree 4.
  
  
  3.3 Example 3: The "Teardrop" orbifold
  
  You  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
  μ-maps.  We  take the isotropy at the top to be C_2. The triangulation looks
  like this, with the gluing being at [1,3].
  
  
       3=====1=====3
      / \   / \   / \
     /   \ /   \ /   \
    5-----2-----4-----5
           \   /
            \ /
             5
  
  
  The source code:
  
    Example  
    gap> M := [ [1,2,3], [1,2,4], [1,3,4], [2,3,5], [2,4,5], [3,4,5] ];;
    gap> iso := rec( 1 := Group( (1,2) ) );;
    gap> mu := [
    >            [ [3], [1,3], [1,2,3], [1,3,4], x -> (1,2) ],
    >            [ [3], [1,3], [1,3,4], [1,2,3], x -> (1,2) ]
    >          ];;
    gap> ot := OrbifoldTriangulation( M, iso, mu, "Teardrop" );
    <OrbifoldTriangulation "Teardrop" of dimension 2. 6 simplices on 5 vertices wi\
    th Isotropy on 1 vertex and nontrivial mu-maps>
    gap> ss := SimplicialSet( ot );
    <The simplicial set of the orbifold triangulation "Teardrop", computed up to d\
    imension 0 with Length vector [ 6 ]>
    gap> R := HomalgRingOfIntegers();
    Z
    gap> c := CreateCoboundaryMatrices( ss, 6, R );;
    gap> C := Cohomology( c, R );
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  Z^(1 x 1)
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  Z/< 2 >
    ----------------------------------------------->>>>  0
    ----------------------------------------------->>>>  Z/< 2 >
    <A graded cohomology object consisting of 7 left modules at degrees
    [ 0 .. 6 ]>
  
  
  This is the Teardrop cohomology.