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.
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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 $
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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 realization 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.
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The resulting spectral sequence is:
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The source of this document is in SpectralSequences.m2:1709:0.