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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $p \\colon E \\to S^n$ be a [[Serre fibration|PB9JXQ2KC3]] over the [[sphere|T62369RFH6]] $S^n$, with fibre $F = p^{-1}(x_0)$ over a point $x_0 \\in S^n$.\nThen there is a long [[exact sequence|JB7ZBFBA9C]] in integral [[singular homology|M0Q5CFJ7ZH]]\n$$\\cdots \\to H_k(F) \\xrightarrow{\\ i_*\\ } H_k(E) \\to H_{k-n}(F) \\to H_{k-1}(F) \\xrightarrow{\\ i_*\\ } H_{k-1}(E) \\to \\cdots,$$\nwhere $i \\colon F \\to E$ is the inclusion of the fibre.\n\n## Remarks\n\nThe base $S^n$ is simply connected because $n \\ge 2$, so the [[Serre spectral sequence|AHW4B5480K]] has $E^2_{s,t} = H_s(S^n; H_t(F))$.\nThis is nonzero only for $s = 0$ and $s = n$, so the only possible differential is $d^n \\colon E^n_{n,t} \\to E^n_{0,t+n-1}$.\nUnfolding this two-column spectral sequence gives the long exact sequence, and the map $H_{k-n}(F) \\to H_{k-1}(F)$ is this differential.\nA dual sequence holds in cohomology.\n\nExample: the [[path space fibration|D3BN7MBKTR]] $\\Omega S^n \\to P S^n \\to S^n$ has contractible total space, where $\\Omega S^n$ is the [[loop space|2JFF5F3EGF]].\nExactness then gives isomorphisms $H_{k-n}(\\Omega S^n) \\cong H_{k-1}(\\Omega S^n)$ for $k \\ge 2$.\nSince $H_0(\\Omega S^n) = \\Z$, one gets $H_j(\\Omega S^n) \\cong \\Z$ when $j$ is a multiple of $n - 1$ and $H_j(\\Omega S^n) = 0$ otherwise.\n\n## Explanations\n\n### L1 human\n\nA fibration over a sphere is a space that looks locally like a product of a piece of sphere with a fixed fibre, but may be twisted globally.\nThe Wang sequence links the homology of the total space with the homology of the fibre.\nAll the twisting is concentrated in a single map from the homology of the fibre to itself, which lowers the degree by $n - 1$.\n\n### L2 human\n\nThe Wang sequence is the fibration analogue of the Mayer–Vietoris sequence, obtained by cutting the sphere into two discs.\nIt is the base-sphere counterpart of the Gysin sequence, which treats fibrations whose fibre is a sphere.\nIts classical use is the computation of the homology of loop spaces of spheres, as in the example above.\n",
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