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SpectralSequences :: Computing the Serre Spectral Sequence associated to a Hopf Fibration

Computing the Serre Spectral Sequence associated to a Hopf Fibration

We compute the Serre Spectral Sequence associated to the Hopf Fibration S1 →S3 →S2. This example is made possible by the minimal triangualtion of this fibration given in the paper "A minimal triangulation of the Hopf map and its application" by K.V. Madahar and K.S Sarkaria. Geom Dedicata, 2000.

We first make the relavant simplicial complexes described on page 110 of the paper. The simplical complex S3 below is a triangualtion of S3.

i1 : B = QQ[a_0..a_2,b_0..b_2,c_0..c_2,d_0..d_2];
i2 : l1 = {a_0*b_0*b_1*c_1,a_0*b_0*c_0*c_1,a_0*a_1*b_1*c_1,b_0*b_1*c_1*d_1,b_0*c_0*c_1*d_2,a_0*a_1*c_1*d_2,a_0*c_0*c_1*d_2,b_0*c_1*d_1*d_2};
i3 : l2 = {b_1*c_1*c_2*a_2,b_1*c_1*a_1*a_2,b_1*b_2*c_2*a_2,c_1*c_2*a_2*d_1,c_1*a_1*a_2*d_2,b_1*b_2*a_2*d_2,b_1*a_1*a_2*d_2,c_1*a_2*d_1*d_2};
i4 : l3 = {c_2*a_2*a_0*b_0,c_2*a_2*b_2*b_0,c_2*c_0*a_0*b_0,a_2*a_0*b_0*d_1,a_2*b_2*b_0*d_2,c_2*c_0*b_0*d_2,c_2*b_2*b_0*d_2,a_2*b_0*d_1*d_2};
i5 : l4 = {a_0*b_0*b_1*d_1,a_0*b_1*d_0*d_1,b_1*c_1*c_2*d_1,b_1*c_2*d_0*d_1,a_0*a_2*c_2*d_1,a_0*c_2*d_0*d_1};
i6 : l5 = {a_0*b_1*d_0*d_2,a_0*a_1*b_1*d_2,b_1*c_2*d_0*d_2,b_1*b_2*c_2*d_2,a_0*c_2*d_0*d_2,a_0*c_0*c_2*d_2};
i7 : S3 = simplicialComplex(join(l1,l2,l3,l4,l5));

We identify the two sphere S2 with the simplical complex S2 defined by the facets {abc, abd, bcd, acd }. The Hopf fibration S1 →S3 →S2 is then realized by the simplicial map p: S3 →S2 defined by ai ↦a, bi ↦b, ci ↦c, and di ↦d.

We now explain how to construct the filtration of S3 obtained by considering the k-sketeltons of this fibration.

The simplical complex F1S3 below is the subsimplical complex of S3 obtained by considering the inverse images of the 1-dimensional faces of the simplical complex S2. We first describe the simplical complex F1S3 in pieces.

For example, to compute f1l1 below, we observe that the inverse image of ab under p is a0b0b1, a0a1b1 etc. All of these inverse images have been computed by hand previously.

i8 : f1l1 = {a_0*b_0*b_1,a_0*a_1*b_1,a_0*c_0*c_1,a_0*a_1*c_1,a_0*a_1*d_2,d_1*d_2,b_0*b_1*c_1,b_0*c_0*c_1,b_0*b_1*d_1,b_0*d_1*d_2,c_1*d_1*d_2,c_0*c_1*d_2};
i9 : f1l2 = {b_1*a_1*a_2,b_1*b_2*a_2,c_1*c_2*a_2,c_1*a_1*a_2,a_1*a_2*d_2,a_2*d_1*d_2,b_1*c_1*c_2,b_1*b_2*c_2,b_1*b_2*d_2,d_1*d_2,c_1*d_1*d_2,c_1*c_2*d_1};
i10 : f1l3 = {a_2*a_0*b_0,a_2*b_2*b_0, c_2*a_2*a_0,c_2*c_0*a_0,a_2*a_0*d_1,a_2*d_1*d_2,b_2*b_0*c_2,c_2*c_0*b_0,b_2*b_0*d_2,b_0*d_1*d_2,c_2*c_0*d_2,d_1*d_2};
i11 : f1l4 = {a_0*b_0*b_1,a_0*a_2,a_0*a_2*c_2,c_1*c_2,a_0*d_0*d_1,a_0*a_2*d_1,b_1*c_1*c_2,b_0*b_1,b_0*b_1*d_1,b_1*d_0*d_1,c_1*c_2*d_1,c_2*d_0*d_1}

o11 = {a b b , a a , a a c , c c , a d d , a a d , b c c , b b , b b d ,
        0 0 1   0 2   0 2 2   1 2   0 0 1   0 2 1   1 1 2   0 1   0 1 1 
      -----------------------------------------------------------------------
      b d d , c c d , c d d }
       1 0 1   1 2 1   2 0 1

o11 : List
i12 : f1l5 = {a_0*a_1*b_1,b_1*b_2,a_0*c_0*c_2,a_0*a_1,a_0*d_0*d_2,a_0*a_1*d_2,b_1*b_2*c_2,c_0*c_2,b_1*d_0*d_2,b_1*b_2*d_2,c_2*d_0*d_2,c_0*c_2*d_2};
i13 : F1S3 = simplicialComplex(join(f1l1,f1l2,f1l3,f1l4,f1l5));

The simplical complex F0S3 below is the subsimplical complex of F1S3 obtained by considering the inverse images of the 0-dimensional faces of the simplical complex S2. Again we describe this simplical complex in pieces.

i14 : f0l1 = {a_0*a_1,b_0*b_1,c_0*c_1,d_1*d_2};
i15 : f0l2 = {a_1*a_2,b_1*b_2,c_1*c_2,d_1*d_2};
i16 : f0l3 = {a_0*a_2,b_0*b_2,c_0*c_2,d_1*d_2};
i17 : f0l4 = {a_0*a_2,b_0*b_1,c_1*c_2,d_0*d_1};
i18 : f0l5 = {a_0*a_1,b_1*b_2,c_0*c_2,d_0*d_2};
i19 : F0S3 = simplicialComplex(join(f0l1,f0l2,f0l3,f0l4,f0l5));

The simplical complex S3 is obtained by considering the inverse images of the 2 dimensional faces of S2.

To compute a simplical version of the Serre spectral sequence for the S1 →S3 →S2 correctly, meaning that the spectral sequence takes the form E2p,q = Hp(S2,Hq(S1,QQ)), we need to use non-reduced homology.

i20 : K = filteredComplex({S3,F1S3,F0S3}, ReducedHomology => false);

We now compute the various pages of the spectral sequence. To make the output intelliagble we prune the spectral sequence.

i21 : E = prune spectralSequence K;
i22 : E0 = E^0

      +------+------+------+
      |  12  |  36  |  36  |
o22 = |QQ    |QQ    |QQ    |
      |      |      |      |
      |{0, 1}|{1, 1}|{2, 1}|
      +------+------+------+
      |  12  |  36  |  36  |
      |QQ    |QQ    |QQ    |
      |      |      |      |
      |{0, 0}|{1, 0}|{2, 0}|
      +------+------+------+

o22 : SpectralSequencePage

Here are the maps.

i23 : E0.dd

o23 = {-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, 4}
                     0

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

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

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

                           36
      {2, -1} : 0 <----- QQ   : {2, 0}
                     0

                 36                                                                                                                        36
      {2, 0} : QQ   <------------------------------------------------------------------------------------------------------------------- QQ   : {2, 1}
                       | 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  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  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  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  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  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  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  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  0  0  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  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  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  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  -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  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 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  -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  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  1  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  -1 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  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  0  0  0  -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 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  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  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  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  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  0  0  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  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  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  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  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  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  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  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  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  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  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  0  0  0  0  0  0  0  0  0  0  0  0  -1 0  0  0  -1 0  0  0  |

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

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

                           36
      {1, -1} : 0 <----- QQ   : {1, 0}
                     0

                 36                                                                                                                        36
      {1, 0} : QQ   <------------------------------------------------------------------------------------------------------------------- QQ   : {1, 1}
                       | 0  0  0  1  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  |
                       | 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  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  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  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  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  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  0  0  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  0  0  0  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  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  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  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  -1 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  0  -1 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  0  0  0  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  0  0  -1 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  -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  -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  0  -1 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  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  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  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  0  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  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  0  0  0  0  0  0  0  0  0  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  0  0  0  0  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  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  0  0  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  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  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  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  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  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  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  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  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  0  0  0  0  0  0  0  0  0  0  0  0  -1 0  0  0  1  |

                 36
      {1, 1} : QQ   <----- 0 : {1, 2}
                       0

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

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

                 12                                                12
      {0, 0} : QQ   <------------------------------------------- QQ   : {0, 1}
                       | 1  1  0  0  0  0  0  0  0  0  0  0  |
                       | -1 0  1  0  0  0  0  0  0  0  0  0  |
                       | 0  -1 -1 0  0  0  0  0  0  0  0  0  |
                       | 0  0  0  1  1  0  0  0  0  0  0  0  |
                       | 0  0  0  -1 0  1  0  0  0  0  0  0  |
                       | 0  0  0  0  -1 -1 0  0  0  0  0  0  |
                       | 0  0  0  0  0  0  1  1  0  0  0  0  |
                       | 0  0  0  0  0  0  -1 0  1  0  0  0  |
                       | 0  0  0  0  0  0  0  -1 -1 0  0  0  |
                       | 0  0  0  0  0  0  0  0  0  1  1  0  |
                       | 0  0  0  0  0  0  0  0  0  -1 0  1  |
                       | 0  0  0  0  0  0  0  0  0  0  -1 -1 |

                 12
      {0, 1} : QQ   <----- 0 : {0, 2}
                       0

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

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

o23 : SpectralSequencePageMap

Now try the E1 page.

i24 : E1 = E^1

      +------+------+------+
      |  4   |  6   |  4   |
o24 = |QQ    |QQ    |QQ    |
      |      |      |      |
      |{0, 1}|{1, 1}|{2, 1}|
      +------+------+------+
      |  4   |  6   |  4   |
      |QQ    |QQ    |QQ    |
      |      |      |      |
      |{0, 0}|{1, 0}|{2, 0}|
      +------+------+------+

o24 : SpectralSequencePage

Here are the maps.

i25 : E1.dd

o25 = {-2, 1} : 0 <----- 0 : {-1, 1}
                     0

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

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

      {-2, 4} : 0 <----- 0 : {-1, 4}
                     0

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

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

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

                 6                        4
      {1, 0} : QQ  <------------------- QQ  : {2, 0}
                      | -1 -1 0  0  |
                      | 1  0  -1 0  |
                      | 0  1  1  0  |
                      | -1 0  0  -1 |
                      | 0  -1 0  1  |
                      | 0  0  -1 -1 |

                 6                      4
      {1, 1} : QQ  <----------------- QQ  : {2, 1}
                      | 1 -1 0 0  |
                      | 0 1  1 0  |
                      | 1 0  1 0  |
                      | 0 -1 0 -1 |
                      | 1 0  0 1  |
                      | 0 0  1 -1 |

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

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

                 4                              6
      {0, 0} : QQ  <------------------------- QQ  : {1, 0}
                      | 1  1  1  0  0  0  |
                      | -1 0  0  1  1  0  |
                      | 0  -1 0  -1 0  1  |
                      | 0  0  -1 0  -1 -1 |

                 4                            6
      {0, 1} : QQ  <----------------------- QQ  : {1, 1}
                      | 1  1  -1 0  0 0 |
                      | -1 0  0  1  1 0 |
                      | 0  -1 0  -1 0 1 |
                      | 0  0  -1 0  1 1 |

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

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

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

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

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

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

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

o25 : SpectralSequencePageMap

Now try the E2 page.

i26 : E2 = E^2

      +------+------+------+
      |  1   |      |  1   |
o26 = |QQ    |0     |QQ    |
      |      |      |      |
      |{0, 1}|{1, 1}|{2, 1}|
      +------+------+------+
      |  1   |      |  1   |
      |QQ    |0     |QQ    |
      |      |      |      |
      |{0, 0}|{1, 0}|{2, 0}|
      +------+------+------+

o26 : SpectralSequencePage

Here are the maps.

i27 : E2.dd

o27 = {-3, 2} : 0 <----- 0 : {-1, 1}
                     0

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

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

      {-3, 5} : 0 <----- 0 : {-1, 4}
                     0

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

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

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

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

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

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

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

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

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

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

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

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

o27 : SpectralSequencePageMap

Note that the modules on the E2 page appear to have been computed correctly. The statement of the Serre spectral sequence, see for example Theorem 1.3 p. 8 of Hatcher’s Spectral Sequence book, asserts that E2p,q = Hp(S2,Hq(S1,QQ)). This is exactly what we obtained above. Also the maps on the E2 page also seem to be computed correctly as the spectral sequence will abut to the homology of S3.

i28 : E3 = E^3

      +------+------+------+
      |      |      |  1   |
o28 = |0     |0     |QQ    |
      |      |      |      |
      |{0, 1}|{1, 1}|{2, 1}|
      +------+------+------+
      |  1   |      |      |
      |QQ    |0     |0     |
      |      |      |      |
      |{0, 0}|{1, 0}|{2, 0}|
      +------+------+------+

o28 : SpectralSequencePage
i29 : E3.dd

o29 = {-4, 3} : 0 <----- 0 : {-1, 1}
                     0

      {-4, 4} : 0 <----- 0 : {-1, 2}
                     0

      {-4, 5} : 0 <----- 0 : {-1, 3}
                     0

      {-4, 6} : 0 <----- 0 : {-1, 4}
                     0

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

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

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

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

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

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

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

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

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

      {-2, 4} : 0 <----- 0 : {1, 2}
                     0

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

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

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

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

      {-3, 5} : 0 <----- 0 : {0, 3}
                     0

      {-4, 2} : 0 <----- 0 : {-1, 0}
                     0

o29 : SpectralSequencePageMap

Thus the E3 page appears to have been computed correctly.