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Hairy ball theorem

A failed attempt to comb a hairy 3-ball (2-sphere), leaving a tuft at each pole
A hairy doughnut (2-torus), on the other hand, is quite easily combable.
A continuous tangent vector field on a 2-sphere with only one pole, in this case a dipole field with index 2. See also an animated version of this graphic.
A hair whorl

The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe)[1] states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres.[2][3] For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in ℝ3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0).

The theorem was first proved by Henri Poincaré for the 2-sphere in 1885,[4] and extended to higher even dimensions in 1912 by Luitzen Egbertus Jan Brouwer.[5]

The theorem has been expressed colloquially as "you can't comb a hairy ball flat without creating a cowlick" or "you can't comb the hair on a coconut".[6]

  1. ^ Renteln, Paul (2013). Manifolds, Tensors, and Forms: An Introduction for Mathematicians and Physicists. Cambridge Univ. Press. p. 253. ISBN 978-1107659698.
  2. ^ Burns, Keith; Gidea, Marian (2005). Differential Geometry and Topology: With a View to Dynamical Systems. CRC Press. p. 77. ISBN 1584882530.
  3. ^ Schwartz, Richard Evan (2011). Mostly Surfaces. American Mathematical Society. pp. 113–114. ISBN 978-0821853689.
  4. ^ Poincaré, H. (1885), "Sur les courbes définies par les équations différentielles", Journal de Mathématiques Pures et Appliquées, 4: 167–244
  5. ^ Georg-August-Universität Göttingen Archived 2006-05-26 at the Wayback Machine - L.E.J. Brouwer. Über Abbildung von Mannigfaltigkeiten / Mathematische Annalen (1912) Volume: 71, page 97-115; ISSN: 0025-5831; 1432-1807/e, full text
  6. ^ Richeson, David S. (23 July 2019). Euler's gem : the polyhedron formula and the birth of topology (New Princeton science library ed.). Princeton. p. 5. ISBN 978-0691191997.{{cite book}}: CS1 maint: location missing publisher (link)
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