{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $M$ be a [[smooth manifold|F45VQ47GPV]] of dimension $n$, Hausdorff and second countable.\nThen there is a smooth [[immersion|6RW68JGBHC]] of $M$ into the [[Euclidean space|0XPR1E85W0]] $\\R^{2n-1}$.\n\n## Remarks\n\nThe weak version gives an immersion of any smooth $n$-manifold, for every $n \\ge 1$, into $\\R^{2n}$, and shows that any smooth map $M \\to \\R^{2n}$ can be approximated by immersions.\nIt follows from a general position argument: the maps whose differential drops rank form a set of codimension $2n - n + 1 = n + 1 \u003e n$ in the space of $1$-jets, so a generic map avoids them.\nThe hypothesis $n \\ge 2$ is needed: the circle $S^1$ admits no immersion into $\\R^1$.\nSuch an immersion would be a local diffeomorphism, so its image would be open and compact in $\\R$, which is impossible.\nExample: the real projective plane $\\R P^2$ immerses in $\\R^3$, as Boy's surface shows.\nFor compact $M$, Ralph Cohen proved the sharper bound $\\R^{2n - \\alpha(n)}$, where $\\alpha(n)$ is the number of ones in the binary expansion of $n$.\n\n## Explanations\n\n### L1 human\n\nAn immersion may cross itself, so it needs less room than an embedding.\nWhitney showed that every $n$-dimensional manifold with $n \\ge 2$ can be drawn without corners or folds, but possibly with self-crossings, in Euclidean space of dimension $2n - 1$.\nFor surfaces this means $\\R^3$: even the Klein bottle and the projective plane can be drawn there with self-crossings.\n\n### L2 human\n\nCompare with the Whitney embedding theorem, which needs dimension $2n$ to avoid self-crossings.\nThe easy bound $2n$ comes from transversality of the $1$-jet of a map to the set of rank-deficient jets.\nThe step down to $2n - 1$ needs a finer study of the singularities of a generic map into $\\R^{2n-1}$.\nThe Smale–Hirsch theorem later reduced the existence of immersions to a question about vector bundles.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "6RW68JGBHC"
    ],
    "kind": "statement",
    "msc": [
      "57R42",
      "58A05"
    ],
    "name": "Whitney immersion theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7996769",
      "wikipedia": "Whitney immersion theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Whitney_immersion_theorem"
      }
    ],
    "tag": "SNZTKKN3GP",
    "uses": [
      "F45VQ47GPV",
      "6RW68JGBHC",
      "0XPR1E85W0"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/SNZTKKN3GP/",
  "links": [
    {
      "field": "about",
      "target": "6RW68JGBHC"
    },
    {
      "field": "uses",
      "target": "F45VQ47GPV"
    },
    {
      "field": "uses",
      "target": "6RW68JGBHC"
    },
    {
      "field": "uses",
      "target": "0XPR1E85W0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/SNZTKKN3GP.md",
  "msc": [
    "57R42",
    "58A05"
  ],
  "name": "Whitney immersion theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/SNZ/SNZTKKN3GP.md",
  "status": null,
  "tag": "SNZTKKN3GP",
  "uri": "m2://SNZTKKN3GP",
  "url": "https://math2.tamnd.com/wiki/whitney-immersion-theorem/",
  "writing": {
    "tag": "SNZTKKN3GP",
    "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
      }
    ]
  }
}