{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nThis defines Hamiltonian Floer homology with coefficients in $\\F_2$ in the simplest setting.\nLet $(M, \\omega)$ be a closed [[symplectic manifold|CAN490EPRT]] such that $\\int_{S^2} v^* \\omega = 0$ for every smooth map $v \\colon S^2 \\to M$.\nLet $H \\colon (\\R / \\Z) \\times M \\to \\R$ be smooth, write $H_t = H(t, \\cdot)$, and let $X_{H_t}$ be the [[Hamiltonian vector field|C5EH3B4BEP]] of $H_t$.\n\n1. Let $\\mathcal{P}(H)$ be the set of contractible loops $x \\colon \\R / \\Z \\to M$ with $x'(t) = X_{H_t}(x(t))$, the contractible $1$-[[periodic orbits|0AA97XA70K]].\nAssume that each of them is nondegenerate: the linearized time-$1$ flow at $x(0)$ does not have $1$ as an eigenvalue.\n2. Let $J = (J_t)_{t \\in \\R / \\Z}$ be a family of [[almost complex structures|9SYWRGVTRS]] compatible with $\\omega$.\nFor $x, y \\in \\mathcal{P}(H)$, let $\\mathcal{M}(x, y)$ be the set of smooth maps $u \\colon \\R \\times (\\R / \\Z) \\to M$ with\n$$\\partial_s u + J_t(u) \\left( \\partial_t u - X_{H_t}(u) \\right) = 0, \\qquad \\lim_{s \\to -\\infty} u(s, t) = x(t), \\qquad \\lim_{s \\to +\\infty} u(s, t) = y(t),$$\non which $\\R$ acts by translation in $s$.\n3. For generic $J$, each set $\\mathcal{M}(x, y)$ is a smooth manifold. When it has dimension $1$, let $n(x, y) \\in \\F_2$ be the number modulo $2$ of elements of $\\mathcal{M}(x, y) / \\R$. Otherwise let $n(x, y) = 0$.\n4. Let $CF(H)$ be the $\\F_2$-vector space with basis $\\mathcal{P}(H)$, and let $\\partial \\colon CF(H) \\to CF(H)$ be the linear map with $\\partial x = \\sum_{y} n(x, y)\\, y$.\n\nThen $\\partial \\circ \\partial = 0$, so $(CF(H), \\partial)$ is a chain complex.\nThe *Floer homology* of $(H, J)$ is $HF(H, J) = \\ker \\partial / \\im \\partial$.\n\n## Remarks\n\nThe solutions $u$ in clause 2 are the negative gradient lines of the action functional on contractible loops, for the $L^2$ metric defined by $J$.\nThey are [[pseudoholomorphic curves|N469E47V75]] perturbed by $H$.\nThat $n(x, y)$ is finite and that $\\partial^2 = 0$ are theorems of Floer, which rest on compactness of the moduli spaces; the assumption that $\\omega = 0$ on spheres rules out sphere bubbling.\nThe homology does not depend on $(H, J)$ up to isomorphism, and Floer proved that it is isomorphic to the singular homology $H_*(M; \\F_2)$.\nSince $\\dim CF(H)$ is the number of contractible $1$-periodic orbits, this gives the Arnold conjecture lower bound in this setting.\nThe complex can be graded by the Conley–Zehnder index when the first Chern class also vanishes on spheres.\n\nExample: let $H$ be a $C^2$-small Morse function $f$ independent of $t$.\nThen the $1$-periodic orbits are the constant loops at critical points of $f$, and the $t$-independent Floer cylinders are the gradient lines of $f$.\nIn this case the Floer complex agrees with the Morse complex of $f$.\n\nSigns, and the order of $x$ and $y$, vary with the conventions for $X_H$ and for the action functional.\nLagrangian Floer homology is defined in the same way, with generators the intersection points of two Lagrangian submanifolds and with strips instead of cylinders.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "NCKFPAGHXF",
    "kind": "definition",
    "msc": [
      "53D40"
    ],
    "name": "Floer homology",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Floer_homology"
      }
    ],
    "tag": "FPXZMN0CH5",
    "uses": [
      "CAN490EPRT",
      "C5EH3B4BEP",
      "0AA97XA70K",
      "9SYWRGVTRS",
      "N469E47V75"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/FPXZMN0CH5/",
  "links": [
    {
      "field": "defines",
      "target": "NCKFPAGHXF"
    },
    {
      "field": "uses",
      "target": "CAN490EPRT"
    },
    {
      "field": "uses",
      "target": "C5EH3B4BEP"
    },
    {
      "field": "uses",
      "target": "0AA97XA70K"
    },
    {
      "field": "uses",
      "target": "9SYWRGVTRS"
    },
    {
      "field": "uses",
      "target": "N469E47V75"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FPXZMN0CH5.md",
  "msc": [
    "53D40"
  ],
  "name": "Floer homology",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FPX/FPXZMN0CH5.md",
  "status": null,
  "tag": "FPXZMN0CH5",
  "uri": "m2://FPXZMN0CH5",
  "url": "https://math2.tamnd.com/wiki/floer-homology-definition/",
  "writing": {
    "tag": "FPXZMN0CH5",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}