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  "body": "\n## Statement\n\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary, of dimension $n$, and let $f \\colon M \\to \\R$ be a [[Morse function|KPQ96640ZJ]].\nFor $0 \\le k \\le n$, let $c_k$ be the number of critical points of $f$ of index $k$.\nLet $F$ be a field, and let $b_k = \\dim_F H_k(M; F)$ be the $k$-th [[Betti number|ST71ZV3FES]] of $M$ over $F$, where $H_k$ is [[singular homology|M0Q5CFJ7ZH]].\nThen:\n\n1. (weak Morse inequalities) $c_k \\ge b_k$ for every $k$;\n2. (strong Morse inequalities) $\\sum_{i=0}^{k} (-1)^{k-i} c_i \\ge \\sum_{i=0}^{k} (-1)^{k-i} b_i$ for every $k$;\n3. $\\sum_{k=0}^{n} (-1)^k c_k = \\sum_{k=0}^{n} (-1)^k b_k = \\chi(M)$, where $\\chi(M)$ is the [[Euler characteristic|RNDDWDAFJ5]] of $M$.\n\n## Remarks\n\nThe idea of the proof: as $c$ increases, the sublevel set $\\set{f \\le c}$ changes up to homotopy only when $c$ crosses a critical value.\nAt such a value it changes by attaching one cell of dimension $k$ for each critical point of index $k$.\nSo $M$ is homotopy equivalent to a CW complex with $c_k$ cells of dimension $k$.\nThe inequalities follow from the cellular chain complex of this CW complex.\nThe strong inequalities imply the weak ones, by adding the inequalities for $k$ and $k - 1$.\nItem 3 is the Euler characteristic of the CW complex, computed once from cells and once from homology.\nExample: the height function on an upright torus has $c_0 = 1$, $c_1 = 2$, $c_2 = 1$, and the torus has $b_0 = 1$, $b_1 = 2$, $b_2 = 1$, so all inequalities are equalities.\nConsequence: every Morse function on a closed orientable surface of genus $g$ has at least $2g + 2$ critical points.\n\n## Explanations\n\n### L1 human\n\nA Morse function needs at least as many critical points of each index as the manifold has holes of that dimension.\nFor example, any nice height function on a doughnut has at least one bottom, one top, and two saddles.\nTopology forces critical points to exist.\n\n### L2 human\n\nThe inequalities are the first quantitative link between the analysis of a function and the topology of its domain.\nThey can be used in two directions.\nThey bound the number of critical points of a function, as in Morse's results on geodesics.\nThey also bound Betti numbers, once one constructs a function with few critical points.\nA Morse function for which all inequalities are equalities is called perfect.\nMorse homology refines the inequalities by building a chain complex on the critical points whose homology is $H_*(M)$.\n",
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