= Relative homology of a fibration over a sphere
{title2=$H_i(E,F;\mathbb Z)\cong H_{i-r}(F;\mathbb Z)$}
For a <Serre fibration> over $S^r$, cover the base by a contractible punctured sphere and a contractible ball. <Excision> relative to the preimage of the punctured sphere identifies the relative pair with the product of $F$ and $(D^r,S^{r-1})$ up to <homotopy>. The base pair has a single free <homology> group in degree $r$, so the <Künneth theorem> gives the displayed shift without a torsion term. The relative <long exact sequence> then gives $\cdots\to H_i(F)\to H_i(E)\to H_{i-r}(F)\to H_{i-1}(F)\to\cdots$. The connecting map retains the twisting of the original <Serre fibration>.
Back to article page
