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  "body": "\n## Definition\n\nLet $a \\ge 1$ and $b$ be coprime integers, let $D^2$ be the closed unit disk and let $\\rho \\colon D^2 \\to D^2$ be the rotation by the angle $2\\pi b / a$.\nThe *fibered solid torus* $T(a, b)$ is the quotient of $D^2 \\times [0, 1]$ by $(x, 0) \\sim (\\rho(x), 1)$, decomposed into the circles that are the images of the segments $\\set{x} \\times [0, 1]$.\nIts central circle, the image of $\\set{0} \\times [0, 1]$, is called *regular* if $a = 1$ and *exceptional* if $a \u003e 1$.\n\nA *Seifert fiber space* is a closed [[3-manifold|R20TB2KEQQ]] $M$ with a decomposition of $M$ into disjoint circles, the *fibers*, such that the following holds.\nEach fiber $C$ has a neighborhood $U$ that is a union of fibers.\nThere is a homeomorphism from $U$ to some $T(a, b)$ that maps fibers to fibers and $C$ to the central circle.\n\n## Remarks\n\nExample: for a closed surface $S$, the product $S \\times S^1$, decomposed into the circles $\\set{s} \\times S^1$, is a Seifert fiber space.\nEach fiber has the neighborhood $B \\times S^1$ with $B$ a small disk around $s$, which is $T(1, 0)$.\n\nNon-example: on the $3$-torus $T^3 = \\R^3 / \\Z^3$, take the decomposition into images of lines parallel to $(1, \\sqrt{2}, 0)$.\nThese images are not circles, because $\\sqrt{2}$ is irrational, so the clause that the pieces are circles fails.\n\nIn $T(a, b)$ every fiber other than the central one closes up after $a$ trips around, so a compact Seifert fiber space has only finitely many exceptional fibers.\nVariants of the definition allow boundary, and, for non-orientable manifolds, fibered solid Klein bottles as neighborhoods.\n",
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