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  "body": "\n## Summary\n\nA vanishing cycle is a homology class in a smooth fibre of a holomorphic function, see the [[definition|QEZFT0EM0G]].\nIt shrinks to a point as the fibre approaches a critical fibre.\nNear a nondegenerate critical point it is represented by an embedded sphere of the middle dimension.\nVanishing cycles generate the homology of the [[Milnor fibre|X5HMTTGB62]] of an isolated singularity, and [[Picard–Lefschetz theory|FFYQAVT7MB]] computes the monodromy in terms of them.\nThe idea was later turned into the vanishing cycle functor on sheaves, used in étale cohomology and in the theory of [[perverse sheaves|5WHXRZDZFK]].\n\n## Explanations\n\n### L1 human\n\nConsider the curves $x^2 + y^2 = t$ in $\\C^2$ for small $t$.\nFor $t \\ne 0$ each curve is a smooth tube, and it contains the circle of real points of radius $\\sqrt{t}$ when $t \u003e 0$.\nAs $t$ goes to $0$, this circle shrinks to the origin, where the curve becomes two crossing lines.\nThe circle is the vanishing cycle: a loop that disappears when the fibre becomes singular.\n",
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