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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $f$ be a holomorphic function near $0$ in $\\C^{n+1}$ with an [[isolated singularity|VTGYEES3VS]] at $0$.\nLet $F$ be its Milnor fibre, the fibre of the [[Milnor fibration|KTFG1S2SMG]] of $f$ at $0$.\nLet $\\mu$ be the [[Milnor number|1ZE51P7GRQ]] of $f$ at $0$.\nThen $F$ is [[homotopy equivalent|XZG6MTSWCN]] to a [[wedge sum|5JSWM1HBX2]] of $\\mu$ copies of the [[sphere|P9CM4GQ2T0]] $S^n$.\n\n## Remarks\n\nThe proof has three steps.\nMorse theory shows that $F$ has the homotopy type of a finite CW complex of dimension at most $n$.\nMilnor shows that, for an isolated singularity, $F$ is $(n-1)$-connected.\nA finite CW complex of dimension $n$ that is $(n-1)$-connected is homotopy equivalent to a wedge of $n$-spheres.\nFor $n \\ge 2$ this follows from the Hurewicz and Whitehead theorems, and for $n = 1$ a connected graph is homotopy equivalent to a wedge of circles.\nThe number of spheres is the rank of $H_n(F; \\Z)$, which Milnor identifies with $\\mu$ through the degree of the gradient map.\n\nExample: for $f = z_0^2 + \\cdots + z_n^2$, $\\mu = 1$, and the fibre $\\set{z_0^2 + \\cdots + z_n^2 = \\delta}$ near $0$ is homotopy equivalent to the single sphere $\\set{x \\in \\R^{n+1} : x_0^2 + \\cdots + x_n^2 = \\delta}$ for $\\delta \u003e 0$.\n\nThe hypothesis that the singularity is isolated is needed.\nFor $f(z_0, z_1, z_2) = z_0 z_1$ on $\\C^3$, the Milnor fibre is homotopy equivalent to a circle, not to a wedge of $2$-spheres.\n\n## Explanations\n\n### L1 human\n\nThe Milnor fibre is a smooth model of a singular hypersurface near its singular point.\nThe theorem says that, up to continuous deformation, this model is just a bunch of spheres of the middle dimension glued at one point.\nThe number of spheres is the Milnor number, which can be computed by algebra.\n\n### L2 human\n\nThe theorem shows that the reduced homology of the Milnor fibre is concentrated in degree $n$ and is free of rank $\\mu$.\nSo the monodromy of the [[Milnor fibration|KTFG1S2SMG]] is described by a single $\\mu \\times \\mu$ integer matrix, and the intersection form on $H_n(F; \\Z)$ is a $\\mu \\times \\mu$ integer form.\nA basis of $H_n(F; \\Z)$ can be chosen of [[vanishing cycles|M5EK9AKXNR]], one for each nondegenerate critical point of a generic perturbation of $f$.\nThis is the setting of [[Picard–Lefschetz theory|FFYQAVT7MB]].\n",
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