nLab
sphere

Context

Topology

Manifolds and cobordisms

Contents

Definition

The n-dimensional unit sphere , or simply n-sphere, is the topological space given by the subset of the (n+1)-dimensional Cartesian space n+1 consisting of all points x whose distance from the origin is 1

S n={x: n+1x=1}.S^n = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = 1 \} \,.

The n-dimensional sphere of radius r is

S r n={x: n+1x=r}.S^n_r = \{ x: \mathbb{R}^{n+1} \;|\; \|x\| = r \} .

Topologically, this is equivalent to the unit sphere for r>0, or a point for r=0.

This is naturally also a smooth manifold of dimension n, with the smooth structure induced from the standard sooth structure on ^n.

Infinite dimensional spheres

One can also talk about a sphere in an arbitrary (possibly infinite-dimensional) normed vector space V:

S(V)={x:Vsuch thatx=1}.S(V) = \{ x: V \;\text{such that}\; {\|x\|} = 1 \} .

If a locally convex topological vector space admits a continuous linear injection into a normed vector space, this can be used to define its sphere. If not, one can still define the sphere as a quotient of the space of non-zero vectors under the scalar action of (0,).

Homotopy theorists define S to be the sphere in the (incomplete) normed vector space (traditionally with the l 2 norm) of infinite sequences almost all of whose values are 0, which is the directed colimit of the S n:

S 1S 0S 1S 2S .S^{-1} \hookrightarrow S^0 \hookrightarrow S^1 \hookrightarrow S^2 \hookrightarrow \cdots S^\infty .

In themselves, these provide nothing new to homotopy theory, as they are at least weakly contractible and usually contractible. However, they are a very useful source of big contractible spaces and so are often used as a starting point for making concrete models of classifying spaces.

If the vector space is a shift space, then contractibility is straightforward to prove.

Theorem

Let V be a shift space of some order. Let SV be its sphere (either via a norm or as the quotient of non-zero vectors). Then SV is contractible.

Proof

Let T:VV be a shift map. The idea is to homotop the sphere onto the image of T, and then down to a point.

It is simplest to start with the non-zero vectors, V{0}. As T is injective, it restricts to a map from this space to itself which commutes with the scalar action of (0,). Define a homotopy H:[0,1]×(V{0})V{0} by H t(v)=(1t)v+tTv. It is clear that, assuming it is well-defined, it is a homotopy from the identity to T. To see that it is well-defined, we need to show that H t(v) is never zero. The only place where it could be zero would be on an eigenvector of T, but as T is a shift map then it has none.

As T is a shift map, it is not surjective and so we can pick some v 0 not in its image. Then we define a homotopy G:[0,1]×(V{0})V{0} by G t(v)=(1t)Tv+tv 0. As v 0 is not in the image of T, this is well-defined on V{0}. Combining these two homotopies results in the desired contraction of V{0}.

If V admits a suitable function defining a spherical subset (such as a norm) then we can modify the above to a contraction of the spherical subset simply by dividing out by this function. If not, as the homotopies above all commute with the scalar action of (0,), they descend to the definition of the sphere as the quotient of V{0}.

Properties

Low dimensions

  • The (1)-sphere is the empty space.
  • The 0-sphere is the disjoint union of two points.
  • The 1-sphere is the circle.
  • The 2-sphere is usual sphere from ordinary geometry.

Note that this violates the convention that a 1-foo is a foo; instead the ruling convention being used is that an n-foo has dimension n. One could follow both by saying ‘n-circle’ instead, although this might get confused with the n-torus.

Revised on December 14, 2011 16:43:08 by Urs Schreiber (89.204.130.117)