# The trivial fibration over the sphere with fibers the sphere

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 = simplicialComplex | 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 $\Delta$ 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

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}; 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