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  "body": "\n## Definition\n\nLet $(M, g)$ be a closed [[Riemannian manifold|7CBT4SM83Y]] of dimension $n + 1 \\ge 3$.\nLet $\\mathcal{Z}_n(M; \\Z_2)$ be the space of $n$-dimensional mod $2$ flat cycles in $M$ that are boundaries, with the flat topology.\nThese are the mod $2$ integral [[currents|Z6QXZX1THM]] $T = \\partial U$, where $U$ is an $(n+1)$-dimensional mod $2$ integral current.\nLet $\\mathbf{M}(T)$ be the mass of $T$.\n\n1. A *sweepout* of $M$ is a map $\\Phi \\colon [0, 1] \\to \\mathcal{Z}_n(M; \\Z_2)$ with $\\Phi(0) = \\Phi(1) = 0$, continuous in the flat topology and with no concentration of mass.\nIt must represent the nontrivial class of $\\pi_1(\\mathcal{Z}_n(M; \\Z_2), 0)$.\nThis group is $\\Z_2$, since the Almgren isomorphism identifies it with $H_{n+1}(M; \\Z_2)$.\n2. The *width* of $M$ is\n$$W(M) = \\inf_{\\Phi} \\ \\sup_{t \\in [0, 1]} \\mathbf{M}\\big(\\Phi(t)\\big),$$\nthe infimum over all sweepouts.\n\n*Almgren–Pitts min-max theory* is the theory of such min-max values, and of the analogous values over $k$-parameter families of cycles.\nIts main theorem states that $W(M) \u003e 0$.\nMoreover, $W(M)$ is the area of a closed embedded [[minimal hypersurface|41X0YW2R9N]], counted with integer multiplicities.\nThis hypersurface is smooth when $3 \\le n + 1 \\le 7$.\nIn general its singular set has [[Hausdorff dimension|SSNFTHZ6YJ]] at most $n - 7$.\n\n## Remarks\n\nPitts proved the smoothness for $3 \\le n + 1 \\le 6$, and the case $n + 1 = 7$ and the higher-dimensional statement follow from the regularity theory of Schoen and Simon.\nThe positivity of the width follows from an isoperimetric inequality: cycles of small mass bound regions of small volume, so a sweepout of small mass would be trivial.\n\nExample: for the round unit sphere $S^3$, the level sets of a height function $x_4$ form a sweepout, $t \\mapsto \\partial\\set{x_4 \\le 2t - 1}$, whose largest member is the equatorial $2$-sphere, of area $4\\pi$.\nHere the width equals $4\\pi$.\n\nNon-example: the family $t \\mapsto \\partial B_{r(t)}(p)$ of boundaries of small geodesic balls, with $r(t) = \\varepsilon \\sin \\pi t$ and $\\varepsilon$ small, starts and ends at $0$ but is not a sweepout.\nIt is contractible in $\\mathcal{Z}_n(M; \\Z_2)$ relative to $0$, because the enclosed regions never fill $M$; so it represents the trivial class and fails clause 1.\n",
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