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sphere

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Topology

Manifolds and cobordisms

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Definition

Finite-dimensional spheres

Definition

The nn-dimensional unit sphere , or simply nn-sphere, is the topological space given by the subset of the (n+1)(n+1)-dimensional Cartesian space n+1\mathbb{R}^{n+1} consisting of all points xx whose distance from the origin is 11

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

The nn-dimensional sphere of radius rr is

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

Topologically, this is equivalent (homeomorphic) to the unit sphere for r>0r \gt 0, or a point for r=0r = 0.

This is naturally also a smooth manifold of dimension nn, with the smooth structure induced from the standard sooth structure on \mathbb{R}^n.

Remark

The n-spheres are coset spaces of orthogonal groups

S nO(n+1)/O(n). S^n \simeq O(n+1)/O(n) \,.

For fix a unit vector in n+1\mathbb{R}^{n+1}. Then its orbit under the defining O(n+1)O(n+1)-action on n+1\mathbb{R}^{n+1} is clearly the canonical embedding S n n+1S^n \hookrightarrow \mathbb{R}^{n+1}. But precisely the subgroup of O(n+1)O(n+1) that consists of rotations around the axis formed by that unit vector stabilizes it, and that subgroup is isomorphic to O(n)O(n), hence S nO(n+1)/O(n)S^n \simeq O(n+1)/O(n).

Infinite dimensional spheres

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

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,)(0,\infty).

Homotopy theorists define S S^\infty to be the sphere in the (incomplete) normed vector space (traditionally with the l 2l^2 norm) of infinite sequences almost all of whose values are 00, which is the directed colimit of the S nS^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 VV be a shift space of some order. Let SVS V be its sphere (either via a norm or as the quotient of non-zero vectors). Then SVS V is contractible.

Proof

Let T:VVT \colon V \to V be a shift map. The idea is to homotop the sphere onto the image of TT, and then down to a point.

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

As TT is a shift map, it is not surjective and so we can pick some v 0v_0 not in its image. Then we define a homotopy G:[0,1]×(V{0})V{0}G \colon [0,1] \times (V \setminus \{0\}) \to V \setminus \{0\} by G t(v)=(1t)Tv+tv 0G_t(v) = (1 - t)T v + t v_0. As v 0v_0 is not in the image of TT, this is well-defined on V{0}V \setminus \{0\}. Combining these two homotopies results in the desired contraction of V{0}V \setminus \{0\}.

If VV 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,)(0,\infty), they descend to the definition of the sphere as the quotient of V{0}V \setminus \{0\}.

Properties

Low dimensions

Note that this violates the convention that a 11-foo is a foo; instead the ruling convention being used is that an nn-foo has dimension nn. One could follow both by saying ‘nn-circle’ instead, although this might get confused with the nn-torus.

References

Formalization

Axiomatization of the homotopy type of the 1-sphere (the circle) and the 2-sphere, as higher inductive types, is in

Visualization of the idea of the construction for the 2-sphere is in

Group actions on spheres

Discussion of free actions by finite groups on spheres includes

See also the ADE classification of such actions on the 7-sphere (as discussed there)

Revised on March 25, 2016 07:58:25 by Urs Schreiber (195.37.209.180)