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  "body": "\n## Definition\n\nLet $M$ be a compact [[smooth manifold|F45VQ47GPV]] without boundary, let $f \\colon M \\to \\R$ be a [[Morse function|KPQ96640ZJ]], and let $g$ be a [[Riemannian metric|Q61VFHCH2J]] on $M$.\nLet $p$ be a critical point, and consider the flow lines of the vector field $-\\nabla f$.\nThe *unstable manifold* $W^u(p)$ is the set of points whose flow line tends to $p$ as time goes to $-\\infty$.\nThe *stable manifold* $W^s(p)$ is the set of points whose flow line tends to $p$ as time goes to $+\\infty$.\nAssume that the pair $(f, g)$ is *Morse–Smale*: $W^u(p)$ and $W^s(q)$ are [[transverse|A2PDZ8GA9C]] for all critical points $p, q$.\nFor $k \\ge 0$, let $C_k$ be the $\\Z/2$-vector space with basis the critical points of $f$ of index $k$.\nLet $p$ be a critical point of index $k$ and $q$ one of index $k - 1$.\nLet $n(p, q) \\in \\Z/2$ be the number, modulo $2$, of flow lines of $-\\nabla f$ from $p$ to $q$, counted up to time shift; it is finite.\nLet $\\partial \\colon C_k \\to C_{k-1}$ be the linear map with $\\partial p = \\sum_q n(p, q) \\, q$.\nThen $\\partial \\circ \\partial = 0$, and the *Morse homology* of $(f, g)$ is the homology of the chain complex $(C_*, \\partial)$.\n\n## Remarks\n\nThe Morse homology is isomorphic to $H_*(M; \\Z/2)$, so it does not depend on $f$ or $g$.\nWith a choice of orientations of the unstable manifolds, one can count flow lines with signs and get a chain complex over $\\Z$ whose homology is $H_*(M; \\Z)$.\nThe identity $\\partial^2 = 0$ comes from broken flow lines $p \\to q \\to r$ with index drop $2$.\nThey are exactly the ends of the one-dimensional spaces of flow lines from $p$ to $r$, and these ends come in pairs.\nExample: the height function on the round sphere $S^n$, $n \\ge 1$, has one critical point of index $0$ and one of index $n$.\nFor $n \\ge 2$ the indices differ by more than one, so $\\partial = 0$.\nFor $n = 1$ there are two flow lines from the maximum to the minimum, so again $\\partial = 0$ modulo $2$.\nIn both cases the homology is $\\Z/2$ in degrees $0$ and $n$.\nNon-example: take a metric on the upright torus in which the flow line leaving the upper saddle runs straight into the lower saddle.\nThen $W^u$ of the upper saddle and $W^s$ of the lower saddle meet along a curve, so they are not transverse.\nThe pair is not Morse–Smale and the definition does not apply; a small change of the metric removes the problem.\n",
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