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Identifying anti-podal points of the two sphere

In this example we compute the spectral sequence arising from the quotient map $\mathbb{S}^2 \rightarrow \mathbb{R} \mathbb{P}^2$, given by identifying anti-podal points. This map can be realized by a simplicial map along the lines of Exercise 27, Section 6.5 of Armstrong's book Basic Topology. In order to give a combinatorial picture of the quotient map $\mathbb{S}^2 \rightarrow \mathbb{R} \mathbb{P}^2$, given by identifying anti-podal points, we first make an appropriate simplicial realization of $\mathbb{S}^2$. Note that we have added a few barycentric coordinates.

i1 : S = ZZ[v1,v2,v3,v4,v5,v6,v15,v12,v36,v34,v46,v25];
 
i2 : twoSphere = simplicialComplex {v3*v4*v5, v5*v4*v15, v15*v34*v4, v15*v34*v1, v34*v1*v6, v34*v46*v6, v36*v46*v6, v3*v4*v46, v4*v46*v34, v3*v46*v36, v1*v6*v2, v6*v2*v36, v2*v36*v12,v36*v12*v3, v12*v3*v5, v12*v5*v25, v25*v5*v15, v2*v12*v25, v1*v2*v25, v1*v25*v15};

We can check that the homology of the simplicial complex twoSphere agrees with that of $\mathbb{S}^2$.

i3 : C = truncate(chainComplex twoSphere,1)

                   12       30       20
o3 = image 0 <-- ZZ   <-- ZZ   <-- ZZ
                                    
     -1          0        1        2

o3 : ChainComplex
 
i4 : prune HH C

o4 = -1 : 0  

            1
      0 : ZZ

      1 : 0  

            1
      2 : ZZ

o4 : GradedModule

We now write down our simplicial complex whose topological realization is $\mathbb{R} \mathbb{P}^2$.

i5 : R = ZZ[a,b,c,d,e,f];
 
i6 : realProjectivePlane = simplicialComplex {a*b*c, b*c*d, c*d*e, a*e*d, e*b*a, e*f*b, d*f*b, a*f*d, c*f*e,a*f*c};

Again we can check that we've entered a simplicial complex whose homology agrees with that of the real projective plane.

i7 : B = truncate(chainComplex realProjectivePlane,1)

                   6       15       10
o7 = image 0 <-- ZZ  <-- ZZ   <-- ZZ
                                   
     -1          0       1        2

o7 : ChainComplex
 
i8 : prune HH B

o8 = -1 : 0             

            1
      0 : ZZ            

      1 : cokernel | 2 |

      2 : 0             

o8 : GradedModule

We now compute the fibers of the anti-podal quotient map $\mathbb{S}^2 \rightarrow \mathbb{R} \mathbb{P}^2$. The way this works for example is: $a = v3 ~ v1, b = v6 ~ v5, d = v36 ~ v15, c = v4 ~ v2, e = v34 ~ v12, f = v46 ~ v25$

The fibers over the vertices of $\mathbb{R} \mathbb{P}^2$ are:

i9 : F0twoSphere = simplicialComplex {v1,v3,v5,v6, v4,v2, v36,v15, v34,v12, v46,v25}

o9 = simplicialComplex | v25 v46 v34 v36 v12 v15 v6 v5 v4 v3 v2 v1 |

o9 : SimplicialComplex

The fibers over the edges of $\mathbb{R}\mathbb{P}^2$ are:

i10 : F1twoSphere = simplicialComplex {v3*v4, v1*v2,v3*v5, v1*v6,v4*v5, v2*v6, v5*v15, v6*v36, v4*v34, v2*v12, v15*v34, v36*v12, v1*v15, v3*v36, v46*v34, v25*v12, v6*v34, v5*v12, v6*v46, v5*v25, v36*v46, v15*v25, v3*v46, v1*v25, v4*v15, v2*v36, v1*v34, v3*v12, v4*v46, v25*v2}

o10 = simplicialComplex | v12v25 v15v25 v5v25 v2v25 v1v25 v34v46 v36v46 v6v46 v4v46 v3v46 v15v34 v6v34 v4v34 v1v34 v12v36 v6v36 v3v36 v2v36 v5v12 v3v12 v2v12 v5v15 v4v15 v1v15 v2v6 v1v6 v4v5 v3v5 v3v4 v1v2 |

o10 : SimplicialComplex

The fibers over the faces is all of $\mathbb{S}^2$.

i11 : F2twoSphere = twoSphere

o11 = simplicialComplex | v5v12v25 v2v12v25 v5v15v25 v1v15v25 v1v2v25 v6v34v46 v4v34v46 v6v36v46 v3v36v46 v3v4v46 v4v15v34 v1v15v34 v1v6v34 v3v12v36 v2v12v36 v2v6v36 v3v5v12 v4v5v15 v1v2v6 v3v4v5 |

o11 : SimplicialComplex

The resulting filtered complex is:

i12 : K = filteredComplex({F2twoSphere, F1twoSphere, F0twoSphere}, ReducedHomology => false)

o12 = -1 : image 0 <-- image 0 <-- image 0 <-- image 0
                                                
           -1          0           1           2

      0 : image 0 <-- image | 1 0 0 0 0 0 0 0 0 0 0 0 | <-- image 0 <-- image 0
                            | 0 1 0 0 0 0 0 0 0 0 0 0 |                  
          -1                | 0 0 1 0 0 0 0 0 0 0 0 0 |     1           2
                            | 0 0 0 1 0 0 0 0 0 0 0 0 |
                            | 0 0 0 0 1 0 0 0 0 0 0 0 |
                            | 0 0 0 0 0 1 0 0 0 0 0 0 |
                            | 0 0 0 0 0 0 1 0 0 0 0 0 |
                            | 0 0 0 0 0 0 0 1 0 0 0 0 |
                            | 0 0 0 0 0 0 0 0 1 0 0 0 |
                            | 0 0 0 0 0 0 0 0 0 1 0 0 |
                            | 0 0 0 0 0 0 0 0 0 0 1 0 |
                            | 0 0 0 0 0 0 0 0 0 0 0 1 |
                       
                      0

      1 : image 0 <-- image | 1 0 0 0 0 0 0 0 0 0 0 0 | <-- image | 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | <-- image 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 0 0 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 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |     2
                            | 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 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                            | 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                      0                                           | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                                                                  | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |
                                                             
                                                            1

                        12       30       20
      2 : image 0 <-- ZZ   <-- ZZ   <-- ZZ
                                         
          -1          0        1        2

o12 : FilteredComplex

We now compute the resulting spectral sequence.

i13 : E = prune spectralSequence K

o13 = E

o13 : SpectralSequence
 
i14 : E^0

      +------+------+------+
      |  12  |  30  |  20  |
o14 = |ZZ    |ZZ    |ZZ    |
      |      |      |      |
      |{0, 0}|{1, 0}|{2, 0}|
      +------+------+------+

o14 : SpectralSequencePage
 
i15 : E^1

      +------+------+------+
      |  12  |  30  |  20  |
o15 = |ZZ    |ZZ    |ZZ    |
      |      |      |      |
      |{0, 0}|{1, 0}|{2, 0}|
      +------+------+------+

o15 : SpectralSequencePage
 
i16 : E^0 .dd

o16 = {-1, 0} : 0 <----- 0 : {-1, 1}
                     0

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

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

      {2, -4} : 0 <----- 0 : {2, -3}
                     0

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

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

                           20
      {2, -1} : 0 <----- ZZ   : {2, 0}
                     0

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

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

                           30
      {1, -1} : 0 <----- ZZ   : {1, 0}
                     0

                 30
      {1, 0} : ZZ   <----- 0 : {1, 1}
                       0

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

                           12
      {0, -1} : 0 <----- ZZ   : {0, 0}
                     0

                 12
      {0, 0} : ZZ   <----- 0 : {0, 1}
                       0

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

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

o16 : SpectralSequencePageMap
 
i17 : E^1 .dd

o17 = {-2, 1} : 0 <----- 0 : {-1, 1}
                     0

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

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

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

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

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

                 30                                                                        20
      {1, 0} : ZZ   <------------------------------------------------------------------- ZZ   : {2, 0}
                       | -1 -1 0  0  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  0  0  -1 -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  0  0  |
                       | 0  1  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                       | -1 0  0  0  0  -1 0  0  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  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                       | 0  -1 0  0  0  0  0  1  0  0  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  0  0  1  0  -1 0  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  0  0  0  1  -1 0  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  -1 0  0  0  0  -1 0  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  0  0  0  1  -1 0  0  0  0  |
                       | 0  0  0  0  0  0  0  0  0  -1 0  0  0  0  0  1  0  0  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  -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  0  1  1  0  0  |
                       | 0  0  0  0  0  -1 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  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  |
                       | 0  0  0  -1 0  0  0  0  0  0  0  0  0  0  -1 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  0  0  0  0  0  -1 0  0  0  0  -1 0  0  0  0  0  0  0  0  |
                       | 0  0  0  0  0  0  0  -1 0  0  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  0  0  -1 0  |
                       | 0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  -1 0  0  0  -1 |

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

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

                 12                                                                                                      30
      {0, 0} : ZZ   <------------------------------------------------------------------------------------------------- ZZ   : {1, 0}
                       | 1  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                       | -1 0  0  0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                       | 0  0  0  0  0  0  0  0  0  1  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  |
                       | 0  0  0  0  0  0  0  0  0  -1 0  0  0  0  1  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0  |
                       | 0  0  0  0  0  0  0  0  0  0  -1 0  0  0  -1 0  0  0  1  1  1  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  1  1  1  0  0  0  0  0  0  |
                       | 0  0  -1 0  0  0  0  0  0  0  0  0  0  0  0  -1 0  0  -1 0  0  0  0  0  1  1  0  0  0  0  |
                       | 0  0  0  0  0  0  -1 0  0  0  0  -1 0  0  0  0  0  0  0  -1 0  0  0  0  0  0  1  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  -1 0  1  0  |
                       | 0  0  0  -1 0  0  0  0  0  0  0  0  0  0  0  0  -1 0  0  0  0  0  -1 0  -1 0  0  0  0  1  |
                       | 0  0  0  0  0  0  0  0  0  0  0  0  0  -1 0  0  0  -1 0  0  0  0  0  -1 0  0  0  0  -1 -1 |
                       | 0  0  0  0  -1 0  0  0  -1 0  0  0  0  0  0  0  0  0  0  0  -1 0  0  0  0  -1 0  -1 0  0  |

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

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

                           12
      {-1, 0} : 0 <----- ZZ   : {0, 0}
                     0

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

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

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

o17 : SpectralSequencePageMap
 
i18 : E^2

      +------+------+------+
      |  1   |      |  1   |
o18 = |ZZ    |0     |ZZ    |
      |      |      |      |
      |{0, 0}|{1, 0}|{2, 0}|
      +------+------+------+

o18 : SpectralSequencePage
 
i19 : E^2 .dd

o19 = {-3, 2} : 0 <----- 0 : {-1, 1}
                     0

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

      {-3, 4} : 0 <----- 0 : {-1, 3}
                     0

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

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

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

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

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

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

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

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

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

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

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

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

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

o19 : SpectralSequencePageMap