{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $M$ and $N$ be [[smooth manifolds|F45VQ47GPV]] of dimensions $m$ and $n$.\nAssume that $m \u003c n$, or that $m = n$ and no connected component of $M$ is compact.\nLet $\\mathrm{Imm}(M, N)$ be the space of [[immersions|6RW68JGBHC]] $M \\to N$ with the $C^1$ topology.\nLet $\\mathrm{Mono}(TM, TN)$ be the space of continuous bundle maps $TM \\to TN$ between [[tangent bundles|WY6K3P4YCG]] that are injective on each fibre, with the compact-open topology.\nThen the map $\\mathrm{Imm}(M, N) \\to \\mathrm{Mono}(TM, TN)$, $f \\mapsto df$, which sends an immersion to its [[differential|6P9PXQDSEN]], is a weak homotopy equivalence.\nThat is, it induces a bijection on path components and isomorphisms on all [[homotopy groups|P976WYEFMB]].\n\n## Remarks\n\nOn path components the theorem says: two immersions are regularly homotopic, that is homotopic through immersions, if and only if their differentials are homotopic through fibrewise injective bundle maps.\nIt also says that $M$ immerses in $N$ as soon as there is a fibrewise injective bundle map $TM \\to TN$.\nSmale first proved the case of spheres in Euclidean space.\nFor $m \u003c n$, regular homotopy classes of immersions $S^m \\to \\R^n$ correspond to $\\pi_m(V_{n, m})$.\nHere $V_{n, m}$ is the Stiefel manifold of orthonormal $m$-frames in $\\R^n$.\nHirsch extended the result to general $M$ and $N$.\nExample: $\\pi_2(V_{3, 2}) = \\pi_2(SO(3)) = 0$, so all immersions $S^2 \\to \\R^3$ are regularly homotopic, which gives sphere eversion.\nThe hypothesis cannot be dropped: for $m = n = 1$ and $M = S^1$, $N = \\R$, the set $\\mathrm{Mono}(TS^1, T\\R)$ is not empty, but there is no immersion $S^1 \\to \\R$.\n\n## Explanations\n\n### L1 human\n\nCan one manifold be drawn smoothly inside another, possibly with self-crossings?\nTo decide this, one only needs to match up tangent directions continuously, without integrating them into an actual map.\nThe same is true for deciding when two such drawings can be deformed into each other.\nThis turns a hard geometric question into a question of algebraic topology.\n\n### L2 human\n\nThe theorem is the first major instance of the homotopy principle.\nIts proof builds the regular homotopy cell by cell over a handle decomposition.\nIt uses a covering homotopy property of the restriction map to a smaller submanifold.\nThe extra room needed comes from the dimension condition or from the openness of $M$.\nCombined with computations of homotopy groups of Stiefel manifolds, it classifies immersions of spheres, and with obstruction theory it gives existence results for immersions of general manifolds.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "6RW68JGBHC",
      "Y2S9GWG1J1"
    ],
    "kind": "statement",
    "msc": [
      "57R42",
      "58A20"
    ],
    "name": "Smale–Hirsch theorem",
    "role": "theorem",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Immersion_(mathematics)"
      }
    ],
    "tag": "7R8W704VGW",
    "uses": [
      "F45VQ47GPV",
      "6RW68JGBHC",
      "WY6K3P4YCG",
      "6P9PXQDSEN",
      "P976WYEFMB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/7R8W704VGW/",
  "links": [
    {
      "field": "about",
      "target": "6RW68JGBHC"
    },
    {
      "field": "about",
      "target": "Y2S9GWG1J1"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "6RW68JGBHC"
    },
    {
      "field": "uses",
      "target": "WY6K3P4YCG"
    },
    {
      "field": "uses",
      "target": "6P9PXQDSEN"
    },
    {
      "field": "uses",
      "target": "P976WYEFMB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/7R8W704VGW.md",
  "msc": [
    "57R42",
    "58A20"
  ],
  "name": "Smale–Hirsch theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/7R8/7R8W704VGW.md",
  "status": null,
  "tag": "7R8W704VGW",
  "uri": "m2://7R8W704VGW",
  "url": "https://math2.tamnd.com/wiki/smale-hirsch-theorem/",
  "writing": {
    "tag": "7R8W704VGW",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}