= Three-sphere bundle over the four-sphere
{title2=$S^3\longrightarrow S(E)\longrightarrow S^4$}
For an oriented rank-four real <vector bundle> $E\to S^4$, its <sphere bundle> has fibre $S^3$. Let $m=\langle e(E),[S^4]\rangle$ be its <Euler class> evaluated on the orientation class. The <Gysin sequence> gives integral <homology> $\mathbb Z$ in degrees $0,7$, $\mathbb Z/m\mathbb Z$ in degree $3$, and a copy of $\mathbb Z$ in degree $4$ exactly when $m=0$, with all other groups zero. Here $\mathbb Z/0\mathbb Z=\mathbb Z$. A transition function $v\mapsto q^k vq^l$ on the <unit quaternions> has Euler number $k+l$, up to an overall orientation sign: evaluating it at a fixed unit vector gives the power map $q\mapsto q^{k+l}$, whose <mapping degree> is $k+l$. The same homology calculation follows by applying the <Mayer–Vietoris sequence> to the two trivializations over the hemispheres.
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