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SpectralSequences :: The trivial fibration over the sphere with fibers the sphere

The trivial fibration over the sphere with fibers the sphere

In this example we compute the spectral sequence associated to the trivial fibration S1 →S1 x S1 →S1, where the map is given by one of the projections. To give a simplicial realization of this fibration we first make a simplical complex which gives a triangulation of S1 ×S1. The simplicial complex that we construct is the triangulation of the torus given in Figure 6.4 of Armstrong’s book Basic Topology and has 18 facets.

i1 : S = ZZ/101[a00,a10,a20,a01,a11,a21,a02,a12,a22];
i2 : Delta = simplicialComplex {a00*a02*a10, a02*a12*a10, a01*a02*a12, a01*a11*a12, a00*a01*a11, a00*a10*a11, a12*a10*a20, a12*a20*a22, a11*a12*a22, a11*a22*a21, a10*a11*a21, a10*a21*a20, a20*a22*a00, a22*a00*a02, a21*a22*a02, a21*a02*a01, a20*a21*a01, a20*a01*a00}

o2 = | a11a12a22 a20a12a22 a21a02a22 a00a02a22 a11a21a22 a00a20a22 a01a02a12 a10a02a12 a01a11a12 a10a20a12 a01a21a02 a00a10a02 a10a11a21 a20a01a21 a10a20a21 a00a01a11 a00a10a11 a00a20a01 |

o2 : SimplicialComplex

We can check that the homology of the simplicial complex Δ agrees with that of the torus S1 ×S1

i3 : C = truncate(chainComplex Delta,1)

                   ZZ 9       ZZ 27       ZZ 18
o3 = image 0 <-- (---)  <-- (---)   <-- (---)
                  101        101         101
     -1                                  
                 0          1           2

o3 : ChainComplex
i4 : prune HH C

o4 = -1 : 0     

            ZZ 1
      0 : (---)
           101

            ZZ 2
      1 : (---)
           101

            ZZ 1
      2 : (---)
           101

o4 : GradedModule

Let S be the simplical complex with facets {A0 A1, A0 A2, A1 A2}. Then S is a triangulation of S1. The simplicial map π: Δ→S given by π(ai,j) = Ai is a combinatorial relization of the trivial fibration S1 →S1 ×S1 →S1. We now make subsimplical complexes arising from the filtrations of the inverse images of the simplicies.

i5 : F1Delta = Delta;
i6 : F0Delta = simplicialComplex {a00*a01, a01*a02, a00*a02, a10*a11,a11*a12,a10*a12, a21*a20,a21*a22,a20*a22};
i7 : K = filteredComplex({F1Delta, F0Delta}, ReducedHomology => false) ;

The resulting spectral sequence is:

i8 : E = prune spectralSequence K

o8 = E

o8 : SpectralSequence
i9 : E^0

     +------+-------+
     |  ZZ 9|  ZZ 18|
o9 = |(---) |(---)  |
     | 101  | 101   |
     |      |       |
     |{0, 1}|{1, 1} |
     +------+-------+
     |  ZZ 9|  ZZ 18|
     |(---) |(---)  |
     | 101  | 101   |
     |      |       |
     |{0, 0}|{1, 0} |
     +------+-------+

o9 : SpectralSequencePage
i10 : E^0 .dd

o10 = {-1, 0} : 0 <----- 0 : {-1, 1}
                     0

      {-1, 1} : 0 <----- 0 : {-1, 2}
                     0

      {-1, 2} : 0 <----- 0 : {-1, 3}
                     0

      {1, -3} : 0 <----- 0 : {1, -2}
                     0

      {1, -2} : 0 <----- 0 : {1, -1}
                     0

                           ZZ 18
      {1, -1} : 0 <----- (---)   : {1, 0}
                     0    101

                 ZZ 18                                                                  ZZ 18
      {1, 0} : (---)   <------------------------------------------------------------- (---)   : {1, 1}
                101       | -1 -1 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |    101
                          | 0  0  -1 -1 0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                          | 1  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                          | 0  0  0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  |
                          | 0  0  0  0  0  0  -1 -1 0  0  0  0  0  0  0  0  0  0  |
                          | 0  0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0  0  |
                          | 0  -1 0  0  0  0  0  0  0  -1 0  0  0  0  0  0  0  0  |
                          | 0  0  -1 0  0  0  0  0  0  0  -1 0  0  0  0  0  0  0  |
                          | 0  0  0  0  0  0  0  -1 0  0  0  -1 0  0  0  0  0  0  |
                          | 0  0  0  0  -1 0  0  0  0  0  0  0  -1 0  0  0  0  0  |
                          | 0  0  0  0  0  0  0  0  0  0  -1 0  0  -1 0  0  0  0  |
                          | 0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  |
                          | 0  0  0  0  0  0  0  0  -1 0  0  0  0  0  0  -1 0  0  |
                          | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  |
                          | 0  0  0  0  0  0  0  0  0  0  0  0  0  -1 0  0  0  -1 |
                          | 0  0  0  0  0  0  0  0  0  -1 0  0  0  0  -1 0  0  0  |
                          | 0  0  0  0  0  -1 0  0  0  0  0  0  0  0  0  0  0  -1 |
                          | 0  0  0  0  0  0  0  0  0  0  0  -1 0  0  0  0  -1 0  |

      {0, -2} : 0 <----- 0 : {0, -1}
                     0

                           ZZ 9
      {0, -1} : 0 <----- (---)  : {0, 0}
                     0    101

                 ZZ 9                                       ZZ 9
      {0, 0} : (---)  <---------------------------------- (---)  : {0, 1}
                101      | 1  1  0  0  0  0  0  0  0  |    101
                         | 0  0  1  1  0  0  0  0  0  |
                         | 0  0  0  0  1  1  0  0  0  |
                         | -1 0  0  0  0  0  1  0  0  |
                         | 0  0  -1 0  0  0  0  1  0  |
                         | 0  0  0  0  -1 0  0  0  1  |
                         | 0  -1 0  0  0  0  -1 0  0  |
                         | 0  0  0  -1 0  0  0  -1 0  |
                         | 0  0  0  0  0  -1 0  0  -1 |

                 ZZ 9
      {0, 1} : (---)  <----- 0 : {0, 2}
                101      0

      {-1, -1} : 0 <----- 0 : {-1, 0}
                      0

o10 : SpectralSequencePageMap
i11 : E^1

      +------+------+
      |  ZZ 3|  ZZ 3|
o11 = |(---) |(---) |
      | 101  | 101  |
      |      |      |
      |{0, 1}|{1, 1}|
      +------+------+
      |  ZZ 3|  ZZ 3|
      |(---) |(---) |
      | 101  | 101  |
      |      |      |
      |{0, 0}|{1, 0}|
      +------+------+

o11 : SpectralSequencePage
i12 : E^1 .dd

o12 = {-2, 1} : 0 <----- 0 : {-1, 1}
                     0

      {-2, 2} : 0 <----- 0 : {-1, 2}
                     0

      {-2, 3} : 0 <----- 0 : {-1, 3}
                     0

      {0, -2} : 0 <----- 0 : {1, -2}
                     0

      {0, -1} : 0 <----- 0 : {1, -1}
                     0

                 ZZ 3                     ZZ 3
      {0, 0} : (---)  <---------------- (---)  : {1, 0}
                101      | 1  1  0  |    101
                         | -1 0  1  |
                         | 0  -1 -1 |

                 ZZ 3                     ZZ 3
      {0, 1} : (---)  <---------------- (---)  : {1, 1}
                101      | -1 0  -1 |    101
                         | 1  -1 0  |
                         | 0  1  1  |

      {-1, -1} : 0 <----- 0 : {0, -1}
                      0

                           ZZ 3
      {-1, 0} : 0 <----- (---)  : {0, 0}
                     0    101

                           ZZ 3
      {-1, 1} : 0 <----- (---)  : {0, 1}
                     0    101

      {-1, 2} : 0 <----- 0 : {0, 2}
                     0

      {-2, 0} : 0 <----- 0 : {-1, 0}
                     0

o12 : SpectralSequencePageMap
i13 : E^2

      +------+------+
      |  ZZ 1|  ZZ 1|
o13 = |(---) |(---) |
      | 101  | 101  |
      |      |      |
      |{0, 1}|{1, 1}|
      +------+------+
      |  ZZ 1|  ZZ 1|
      |(---) |(---) |
      | 101  | 101  |
      |      |      |
      |{0, 0}|{1, 0}|
      +------+------+

o13 : SpectralSequencePage