We compute the Serre Spectral Sequence associated to the Hopf Fibration $S^1 \rightarrow S^3 \rightarrow S^2$. This example is made possible by the minimal triangulation 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 relevant simplicial complexes described on page 110 of the paper. The simplicial complex $S3$ below is a triangulation of $S^3$.
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We identify the two sphere $S^2$ with the simplicial complex $S2$ defined by the facets $\{abc, abd, bcd, acd \}$. The Hopf fibration $S^1 \rightarrow S^3 \rightarrow S^2$ is then realized by the simplicial map $p: S3 \rightarrow S2$ defined by $a_i \mapsto a$, $b_i \mapsto b$, $c_i \mapsto c$, and $d_i \mapsto d$.
We now explain how to construct the filtration of $S3$ obtained by considering the $k$-skeletons of this fibration.
The simplicial complex $F1S3$ below is the subsimplicial complex of $S3$ obtained by considering the inverse images of the $1$-dimensional faces of the simplicial complex $S2$. We first describe the simplicial complex $F1S3$ in pieces.
For example, to compute $f1l1$ below, we observe that the inverse image of $ab$ under $p$ is $a_0b_0b_1, a_0a_1b_1$ etc. All of these inverse images have been computed by hand previously.
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The simplicial complex $F0S3$ below is the subsimplicial complex of $F1S3$ obtained by considering the inverse images of the $0$-dimensional faces of the simplicial complex $S2$. Again we describe this simplicial complex in pieces.
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The simplicial complex $S3$ is obtained by considering the inverse images of the $2$ dimensional faces of $S2$.
To compute a simplicial version of the Serre spectral sequence for the $S^1 \rightarrow S^3 \rightarrow S^2$ correctly, meaning that the spectral sequence takes the form $E^2_{p,q} = H_p(S^2,H_q(S^1,QQ))$, we need to use non-reduced homology.
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We now compute the various pages of the spectral sequence. To make the output intelligible we prune the spectral sequence.
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Here are the maps.
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Now try the $E^1$ page.
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Here are the maps.
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Now try the $E^2$ page.
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Here are the maps.
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Note that the modules on the $E^2$ 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 $E^2_{p,q} = H_p(S^2,H_q(S^1,QQ))$. This is exactly what we obtained above. Also the maps on the $E^2$ page also seem to be computed correctly as the spectral sequence will abut to the homology of $S^3$.
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Thus the E^3 page appears to have been computed correctly.
The source of this document is in /build/reproducible-path/macaulay2-1.26.05+ds/M2/Macaulay2/packages/SpectralSequences.m2:1906:0.