In this example we compute the spectral sequence associated to the trivial fibration $\mathbb{S}^1 \rightarrow \mathbb{S}^1 x \mathbb{S}^1 \rightarrow \mathbb{S}^1$, where the map is given by one of the projections. To give a simplicial realization of this fibration we first make a simplicial complex which gives a triangulation of $\mathbb{S}^1 \times \mathbb{S}^1$. 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
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We can check that the homology of the simplicial complex $\Delta$ agrees with that of the torus $\mathbb{S}^1 \times \mathbb{S}^1 $
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
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Let $S$ be the simplicial complex with facets $\{A_0 A_1, A_0 A_2, A_1 A_2\}$. Then $S$ is a triangulation of $S^1$. The simplicial map $\pi : \Delta \rightarrow S$ given by $\pi(a_{i,j}) = A_i$ is a combinatorial relization of the trivial fibration $\mathbb{S}^1 \rightarrow \mathbb{S}^1 \times \mathbb{S}^1 \rightarrow \mathbb{S}^1$. We now make subsimplicial 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};
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i7 : K = filteredComplex({F1Delta, F0Delta}, ReducedHomology => false) ;
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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
|