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  "body": "\n## Statement\n\nLet $f \\colon U \\to \\C$ be a [[holomorphic function|507C34W0ZT]] on an open neighbourhood $U$ of $0$ in $\\C^{n+1}$, with $f(0) = 0$ and $f$ not identically zero.\nFor $\\varepsilon \u003e 0$, let $S_\\varepsilon$ be the [[sphere|P9CM4GQ2T0]] of radius $\\varepsilon$ about $0$ and let $K_\\varepsilon = S_\\varepsilon \\cap f^{-1}(0)$, the [[link|C831AN3RFM]] of $f^{-1}(0)$ at $0$.\nThen there is $\\varepsilon_0 \u003e 0$ such that, for every $0 \u003c \\varepsilon \\le \\varepsilon_0$:\n\n1. the map $\\phi = f / \\abs{f} \\colon S_\\varepsilon \\setminus K_\\varepsilon \\to S^1$ is a smooth locally trivial [[fibre bundle|CX4M8K61AB]];\n2. each fibre $F_\\theta = \\phi^{-1}(e^{i\\theta})$ is a smooth manifold of real dimension $2n$, called the *Milnor fibre* of $f$ at $0$.\n\n## Remarks\n\nMilnor proved this in his 1968 book on singular points of complex hypersurfaces.\nThe proof shows, with the curve selection lemma, that for small $\\varepsilon$ the map $\\phi$ has no critical points on $S_\\varepsilon \\setminus K_\\varepsilon$.\nOne then builds a vector field on $S_\\varepsilon \\setminus K_\\varepsilon$ that $\\phi$ sends to the unit tangent field of $S^1$ and that is controlled near $K_\\varepsilon$.\nIts flow carries each fibre onto the others.\nAn equivalent form, used often, is the fibration $f \\colon B_\\varepsilon \\cap f^{-1}(\\partial D_\\delta) \\to \\partial D_\\delta$, where $B_\\varepsilon$ is the closed ball and $D_\\delta$ a disc of radius $0 \u003c \\delta \\ll \\varepsilon$.\n\nExample: for $f(z_0, z_1) = z_0 z_1$ on $\\C^2$, the link is a pair of circles forming a Hopf link in $S^3$.\nThe fibre in the tube form is $\\set{z_0 z_1 = \\delta} \\cap B_\\varepsilon$, an annulus, which has the homotopy type of a circle.\n\nThe radius must be small.\nOn a large sphere, $f^{-1}(0)$ may meet the sphere non-transversally or other critical points of $f$ may intervene, and the conclusion can fail.\n\n## Explanations\n\n### L1 human\n\nTake a hypersurface with a singular point and look at it through a small sphere around that point.\nAway from the hypersurface, the argument of $f$ is an angle, so it gives a map from the rest of the sphere to the circle.\nMilnor's theorem says that this map is a fibration: all its fibres look the same.\nThe sphere minus the link is swept out by them as the angle turns once around the circle.\nThe common fibre, the Milnor fibre, is a smooth model of the hypersurface near the singular point, slightly pushed off the singular value.\n\n### L2 human\n\nThe Milnor fibration is the starting point of the topology of hypersurface singularities.\nIts fibre carries the [[Milnor number|1ZE51P7GRQ]], and for an isolated singularity the [[fibre is a bouquet of spheres|X5HMTTGB62]].\nGoing once around the circle gives a diffeomorphism of the fibre, whose action on homology is the monodromy studied in the [[monodromy theorem|DQP63REKEK]].\nFor plane curves, $n = 1$, the link is a link in $S^3$ and the Milnor fibre is a Seifert surface of it, which connects singularity theory with knot theory.\n",
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