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  "body": "\n## Statement\n\nLet $p \\colon E \\to B$ be a [[Serre fibration|PB9JXQ2KC3]], and let $B$ be [[path-connected|0WB6DH499T]].\nLet $b_0 \\in B$, let $F = p^{-1}(b_0)$ be the fibre, and let $x_0 \\in F$.\nThen there is an [[exact sequence|JB7ZBFBA9C]]\n$$\\cdots \\to \\pi_n(F, x_0) \\xrightarrow{i_*} \\pi_n(E, x_0) \\xrightarrow{p_*} \\pi_n(B, b_0) \\xrightarrow{\\partial} \\pi_{n-1}(F, x_0) \\to \\cdots \\to \\pi_0(F, x_0) \\to \\pi_0(E, x_0) \\to 0.$$\nHere $\\pi_n$ are the [[homotopy groups|P976WYEFMB]], $i \\colon F \\to E$ is the inclusion, and $\\partial$ is a homomorphism for $n \\ge 2$.\nAt the terms $\\pi_0$ and at $\\pi_1(B, b_0)$, exactness is meant as pointed sets: the kernel of a map is the preimage of the base point.\n\n## Remarks\n\nThe key step is that $p_* \\colon \\pi_n(E, F, x_0) \\to \\pi_n(B, b_0)$ is an isomorphism for $n \\ge 1$, proved by lifting maps of disks with the homotopy lifting property.\nThe sequence then comes from the long exact sequence of the pair $(E, F)$ in [[relative homotopy groups|ZHHMDZWA7F]].\nThe map $\\partial$ lifts a based map of a disk into $B$ to $E$ and restricts it to the boundary sphere, which lands in $F$.\nFor the Hopf fibration $S^1 \\to S^3 \\to S^2$, the sequence gives $\\pi_2(S^2) \\cong \\pi_1(S^1) \\cong \\Z$ and $\\pi_n(S^3) \\cong \\pi_n(S^2)$ for $n \\ge 3$.\nFor the covering $\\R \\to S^1$ the fibre $\\Z$ is discrete, so $\\pi_n(S^1) \\cong \\pi_n(\\R) = 0$ for $n \\ge 2$.\n\n## Explanations\n\n### L1 human\n\nA fibration is a space $E$ built over a base $B$ with copies of a fibre $F$ over each point.\nThe sequence says that the homotopy groups of the three spaces are tied together tightly.\nIf two of them are known, the third is often determined.\n\n### L2 human\n\nThis is the most used tool for computing homotopy groups.\nIt gives $\\pi_3(S^2) \\cong \\Z$ from the Hopf fibration, and computes homotopy groups of Lie groups and homogeneous spaces from fibrations such as $SO(n) \\to SO(n+1) \\to S^n$.\nFor the path space fibration $\\Omega B \\to PB \\to B$ with contractible total space, it gives $\\pi_n(\\Omega B) \\cong \\pi_{n+1}(B)$.\nThe analogous sequence of a pair $A \\subseteq X$ has the form $\\cdots \\to \\pi_n(A) \\to \\pi_n(X) \\to \\pi_n(X, A) \\to \\pi_{n-1}(A) \\to \\cdots$.\n",
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