A torus may be brushed smooth but a sphere can not.
Shaggy dog theorem
By Hubert Shutrick
June 19, 2008
The sphere does not admit a continuous unit vector field.
In math folklore the theorem reads:
Consider the unit sphere |v| = 1 in three dimensional space. Let N be the point
A continuous vector field would be given by a continuous unit vector function u(v) such that v and u(v) are orthogonal. If one exists, consider the map
ft(v) = v cos(t π) + u(v) sin(t π).
Consider the sphere as consisting of the semicircular longitudes joining N and S each defined by a vector e on the equator E: it is said to be the suspension of the circle E. Any map g of E to itself can be extended to a map of the sphere to itself by suspending it, that is, each latitude goes to itself in such a way that the point on the longitude e goes to the point on the g(e) longitude. If g winds E n times round itself, then its suspension will wrap the sphere round n times leaving N and S fixed. In the case of the circle E, it is well known that deforming g continuously leaves the integer n fixed so the set of equivalence classes of maps under homotopy from the circle to itself is in 1-1 correspondence with the set of integers. Also, deforming g continuously deforms its suspension continuously. Suspension gives then a mapping m of the integers to the set of equivalence classes of maps of the sphere into itself under homotopy.
To prove that it is injective requires some basic homotopy theory and is usually proved using the Hopf fibration which is a map from the three dimensional sphere onto the sphere such that the inverse image of each point is a circle. In fact the homotopy classes of maps of the sphere to itself form an abelian group and it can be proved that m is a group isomorphism.
This method of proof applies to all even-dimensional n-spheres none of which admit a unit vector field because the antipodal map changes the orientation of the equatorial (n-1)-sphere. Odd dimensional spheres can admit several independent (at each point) unit vector fields. The formula for how many is very elegantly deduced in Husemoler's Fibre Bundles, McGraw-Hill, 1966, using Clifford algebras. They give the vector fields but it was Frank Adams: Ann. Math. 75, 603-632 (1962) who proved that there weren't any others. The 3-sphere admits three independent fields: it is parallelizable. I had the very good fortune to be in Manchester in the sixties when Adams was there and attended the seminars he arranged and got to know him quite well.
Note: In case it would be of interest in itself, here is the Hopf fibration:
The Hopf fibration is defined by considering the 3-sphere as the subspace of C² defined by
|z1|² + |z2|² = 1.
The map takes (z1, z2) to z1/z2 in the complex projective line, which is the 2-sphere. It takes
z'1 = z1exp(iθ) and
z'2 = z2exp(-iθ)
Hence the circle.
tuft (noun) - a cluster of short, fluffy threads, used to decorate cloth, as for a bedspread, robe, bath mat, or window curtain.
whorl (noun) - 1. a circular arrangement of like parts, as leaves or flowers, around a point on an axis; verticil. 2. one of the turns or volutions of a spiral shell. 3. anything shaped like a coil.